Questions S1 - A-Level Maths

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.

No sixes are thrown.
Exactly four sixes are thrown.
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.
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.
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 correct	1	2	3	4	5	6	7
Frequency	1	2	3	3	4	7	5

Draw a vertical line chart to illustrate the data.
State the type of skewness shown by your diagram.
Calculate the mean and the mean squared deviation of the data.
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 $\mathrm { P } ( A ) = \frac { 7 } { 10 }$.

Calculate $\mathrm { P } ( A \cap B )$. The probability that Isobel does not score and her parents do not attend is 0.1 .
Draw a Venn diagram showing the events $A$ and $B$, and mark in the probability corresponding to each of the regions of your diagram.
Are events $A$ and $B$ independent? Give a reason for your answer.
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 $$\mathrm { P } ( X = r ) = k r ( 8 - r ) \text { for } r = 1,2,3,4 .$$

Show that $k = \frac { 1 } { 50 }$.
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.

Calculate the number of ways in which Peter may choose his three-course meal.
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.
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 $\mathrm { B } ( 3,0.87 )$.

Calculate the probability that Douglas hits 20 twice.
Douglas aims fifty sets of 3 darts at the number 20. Find the expected number of sets for which Douglas hits 20 twice.
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 .

Copy and complete the probability tree diagram below, to illustrate the information above.
\includegraphics[max width=\textwidth, alt={}, center]{16488e7a-36fb-47f1-8dbf-dec57387f2bf-4_469_668_861_699} Calculate the probabilities of the following events.
The test gives a positive result.
The test gives a correct result.
The painting is genuine, given a positive result.
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.
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.
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.

Find the probability that
(A) exactly one of the rocks contains fossils,
(B) at least one of the rocks contains fossils.
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 $n$.
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 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.

In how many ways can she select the 4 tracks?
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

Express the data in the form of a grouped frequency table.
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 $$n = 56 , \quad \Sigma x = 3026 , \quad \Sigma x ^ { 2 } = 178890 .$$

Calculate the mean and standard deviation of the marks.
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.
The formula $y = 1.2 x - 10$ is used to scale the marks. Find the mean and standard deviation of the scaled marks.

OCR MEI S1 2007 June Q4

8 marks Easy -1.2

4 A local council has introduced a recycling scheme for aluminium, paper and kitchen waste. 50 residents are asked which of these materials they recycle. The numbers of people who recycle each type of material are shown in the Venn diagram.
\includegraphics[max width=\textwidth, alt={}, center]{5e4f3310-b96e-43db-9b6d-61da3270db06-3_803_803_406_671} One of the residents is selected at random.

Find the probability that this resident recycles
(A) at least one of the materials,
(B) exactly one of the materials.
Given that the resident recycles aluminium, find the probability that this resident does not recycle paper. Two residents are selected at random.
Find the probability that exactly one of them recycles kitchen waste.

OCR MEI S1 2007 June Q5

6 marks Easy -1.8

5 A GCSE geography student is investigating a claim that global warming is causing summers in Britain to have more rainfall. He collects rainfall data from a local weather station for 2001 and 2006. The vertical line chart shows the number of days per week on which some rainfall was recorded during the 22 weeks of summer 2001.
\includegraphics[max width=\textwidth, alt={}, center]{5e4f3310-b96e-43db-9b6d-61da3270db06-4_720_1557_443_296} Number of days per week with rain recorded in summer 2001

Show that the median of the data is 4 , and find the interquartile range.
For summer 2006 the median is 3 and the interquartile range is also 3. The student concludes that the data demonstrate that global warming is causing summer rainfall to decrease rather than increase. Is this a valid conclusion from the data? Give two brief reasons to justify your answer.

OCR MEI S1 2007 June Q6

7 marks Standard +0.3

6 In a phone-in competition run by a local radio station, listeners are given the names of 7 local personalities and are told that 4 of them are in the studio. Competitors phone in and guess which 4 are in the studio.

Show that the probability that a randomly selected competitor guesses all 4 correctly is $\frac { 1 } { 35 }$. Let $X$ represent the number of correct guesses made by a randomly selected competitor. The probability distribution of $X$ is shown in the table.
$r$ 0 1 2 3 4
$\mathrm { P } ( X = r )$ 0 $\frac { 4 } { 35 }$ $\frac { 18 } { 35 }$ $\frac { 12 } { 35 }$ $\frac { 1 } { 35 }$
Find the expectation and variance of $X$.

OCR MEI S1 2007 June Q7

18 marks Standard +0.3

7 A screening test for a particular disease is applied to everyone in a large population. The test classifies people into three groups: 'positive', 'doubtful' and 'negative'. Of the population, $3 \%$ is classified as positive, $6 \%$ as doubtful and the rest negative. In fact, of the people who test positive, only $95 \%$ have the disease. Of the people who test doubtful, $10 \%$ have the disease. Of the people who test negative, $1 \%$ actually have the disease. People who do not have the disease are described as 'clear'.

Copy and complete the tree diagram to show this information.
\includegraphics[max width=\textwidth, alt={}, center]{5e4f3310-b96e-43db-9b6d-61da3270db06-5_830_1157_845_536}
Find the probability that a randomly selected person tests negative and is clear.
Find the probability that a randomly selected person has the disease.
Find the probability that a randomly selected person tests negative given that the person has the disease.
Comment briefly on what your answer to part (iv) indicates about the effectiveness of the screening test. Once the test has been carried out, those people who test doubtful are given a detailed medical examination. If a person has the disease the examination will correctly identify this in $98 \%$ of cases. If a person is clear, the examination will always correctly identify this.
A person is selected at random. Find the probability that this person either tests negative originally or tests doubtful and is then cleared in the detailed medical examination.

OCR MEI S1 2007 June Q8

18 marks Standard +0.3

8 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.

Find the probability that at least 4 of the 17 applicants are successful.
Find the expected number of successful applicants in the sample.
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.
(A) Write down suitable null and alternative hypotheses for the test.
(B) Give a reason for your choice of the alternative hypothesis.
Find the critical region for the test at the $5 \%$ level, showing all of your calculations.
Explain why the critical region found in part (v) would be unaltered if a $10 \%$ significance level were used.

OCR MEI S1 2008 June Q1

6 marks Moderate -0.8

1 In a survey, a sample of 44 fields is selected. Their areas ( $x$ hectares) are summarised in the grouped frequency table.

Area $( x )$	$0 < x \leqslant 3$	$3 < x \leqslant 5$	$5 < x \leqslant 7$	$7 < x \leqslant 10$	$10 < x \leqslant 20$
Frequency	3	8	13	14	6

Calculate an estimate of the sample mean and the sample standard deviation.
Determine whether there could be any outliers at the upper end of the distribution.

OCR MEI S1 2008 June Q2

8 marks Moderate -0.8

2 In the 2001 census, people living in Wales were asked whether or not they could speak Welsh. A resident of Wales is selected at random.

$W$ is the event that this person speaks Welsh.
$C$ is the event that this person is a child.

You are given that $\mathrm { P } ( W ) = 0.20 , \mathrm { P } ( C ) = 0.17$ and $\mathrm { P } ( W \cap C ) = 0.06$.

Determine whether the events $W$ and $C$ are independent.
Draw a Venn diagram, showing the events $W$ and $C$, and fill in the probability corresponding to each region of your diagram.
Find $\mathrm { P } ( W \mid C )$.
Given that $\mathrm { P } \left( W \mid C ^ { \prime } \right) = 0.169$, use this information and your answer to part (iii) to comment very briefly on how the ability to speak Welsh differs between children and adults.

OCR MEI S1 2008 June Q3

7 marks Moderate -0.3

3 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.

$r$	0	1	2	3
$\mathrm { P } ( X = r )$	0.5	0.35	$p$	$q$

(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 $q$.
Find the variance of $X$.

OCR MEI S1 2008 June Q4

7 marks Moderate -0.8

4 A small business has 8 workers. On a given day, the probability that any particular worker is off sick is 0.05 , independently of the other workers.

A day is selected at random. Find the probability that
(A) no workers are off sick,
(B) more than one worker is off sick.
There are 250 working days in a year. Find the expected number of days in the year on which more than one worker is off sick.

OCR MEI S1 2008 June Q5

8 marks Moderate -0.3

5 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.

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.
Carry out the test at the $5 \%$ significance level.

OCR MEI S1 2008 June Q6

16 marks Moderate -0.3

6 In a large town, 79\% of the population were born in England, 20\% in the rest of the UK and the remaining 1\% overseas. Two people are selected at random. You may use the tree diagram below in answering this question.
\includegraphics[max width=\textwidth, alt={}, center]{be764df3-ff20-415d-9c5c-10edabf350de-4_946_1119_580_513}

Find the probability that
(A) both of these people were born in the rest of the UK,
(B) at least one of these people was born in England,
(C) neither of these people was born overseas.
Find the probability that both of these people were born in the rest of the UK given that neither was born overseas.
(A) Five people are selected at random. Find the probability that at least one of them was not born in England.
(B) An interviewer selects $n$ people at random. The interviewer wishes to ensure that the probability that at least one of them was not born in England is more than $90 \%$. Find the least possible value of $n$. You must show working to justify your answer.

OCR MEI S1 2008 June Q7

20 marks Moderate -0.8

7 The histogram shows the age distribution of people living in Inner London in 2001.
\includegraphics[max width=\textwidth, alt={}, center]{be764df3-ff20-415d-9c5c-10edabf350de-5_814_1383_349_379} Data sourced from the 2001 Census, \href{http://www.statistics.gov.uk}{www.statistics.gov.uk}

State the type of skewness shown by the distribution.
Use the histogram to estimate the number of people aged under 25.
The table below shows the cumulative frequency distribution.
Age 20 30 40 50 65 100
Cumulative frequency (thousands) 660 1240 1810 $a$ 2490 2770
(A) Use the histogram to find the value of $a$.
(B) Use the table to calculate an estimate of the median age of these people. The ages of people living in Outer London in 2001 are summarised below.
Age ( $x$ years) $0 \leqslant x < 20$ $20 \leqslant x < 30$ $30 \leqslant x < 40$ $40 \leqslant x < 50$ $50 \leqslant x < 65$ $65 \leqslant x < 100$
Frequency (thousands) 1120 650 770 590 680 610
Illustrate these data by means of a histogram.
Make two brief comments on the differences between the age distributions of the populations of Inner London and Outer London.
The data given in the table for Outer London are used to calculate the following estimates. Mean 38.5, median 35.7, midrange 50, standard deviation 23.7, interquartile range 34.4.
The final group in the table assumes that the maximum age of any resident is 100 years. These estimates are to be recalculated, based on a maximum age of 105, rather than 100. For each of the five estimates, state whether it would increase, decrease or be unchanged.

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.

(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.
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.
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.

\(r\)	0	1	2	3	4
\(\mathrm { P } ( X = r )\)	0	\(\frac { 4 } { 35 }\)	\(\frac { 18 } { 35 }\)	\(\frac { 12 } { 35 }\)	\(\frac { 1 } { 35 }\)

Area \(( x )\)	\(0 < x \leqslant 3\)	\(3 < x \leqslant 5\)	\(5 < x \leqslant 7\)	\(7 < x \leqslant 10\)	\(10 < x \leqslant 20\)
Frequency	3	8	13	14	6

\(r\)	0	1	2	3
\(\mathrm { P } ( X = r )\)	0.5	0.35	\(p\)	\(q\)

Age	20	30	40	50	65	100
Cumulative frequency (thousands)	660	1240	1810	\(a\)	2490	2770

Age ( \(x\) years)	\(0 \leqslant x < 20\)	\(20 \leqslant x < 30\)	\(30 \leqslant x < 40\)	\(40 \leqslant x < 50\)	\(50 \leqslant x < 65\)	\(65 \leqslant x < 100\)
Frequency (thousands)	1120	650	770	590	680	610

Questions S1 (1967 questions)

Browse by module

Browse by board

OCR MEI S1 2005 June Q4

OCR MEI S1 2005 June Q7

OCR MEI S1 2006 June Q1

OCR MEI S1 2006 June Q2

OCR MEI S1 2006 June Q3

OCR MEI S1 2006 June Q4

OCR MEI S1 2006 June Q5

OCR MEI S1 2006 June Q6

OCR MEI S1 2006 June Q7

OCR MEI S1 2007 June Q1

OCR MEI S1 2007 June Q2

OCR MEI S1 2007 June Q3

OCR MEI S1 2007 June Q4

OCR MEI S1 2007 June Q5

OCR MEI S1 2007 June Q6

OCR MEI S1 2007 June Q7

OCR MEI S1 2007 June Q8

OCR MEI S1 2008 June Q1

OCR MEI S1 2008 June Q2

OCR MEI S1 2008 June Q3

OCR MEI S1 2008 June Q4

OCR MEI S1 2008 June Q5

OCR MEI S1 2008 June Q6

OCR MEI S1 2008 June Q7

OCR MEI S1 Q1