Questions S1 (2020 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Sort by: Default | Easiest first | Hardest first
OCR MEI S1 2011 June Q4
7 marks Moderate -0.8
Two fair six-sided dice are thrown. The random variable \(X\) denotes the difference between the scores on the two dice. The table shows the probability distribution of \(X\).
\(r\)012345
P(X = r)\(\frac{1}{6}\)\(\frac{5}{18}\)\(\frac{2}{9}\)\(\frac{1}{6}\)\(\frac{1}{9}\)\(\frac{1}{18}\)
  1. Draw a vertical line chart to illustrate the probability distribution. [2]
  2. Use a probability argument to show that
    1. P(X = 1) = \(\frac{5}{18}\). [2]
    2. P(X = 0) = \(\frac{1}{6}\). [1]
  3. Find the mean value of \(X\). [2]
OCR MEI S1 2011 June Q5
8 marks Moderate -0.8
In a recent survey, a large number of working people were asked whether they worked full-time or part-time, with part-time being defined as less than 25 hours per week. One of the respondents is selected at random. • \(W\) is the event that this person works part-time. • \(F\) is the event that this person is female. You are given that P(\(W\)) = 0.14, P(\(F\)) = 0.41 and P(\(W \cap F\)) = 0.11.
  1. Draw a Venn diagram showing the events \(W\) and \(F\), and fill in the probability corresponding to each of the four regions of your diagram. [3]
  2. Determine whether the events \(W\) and \(F\) are independent. [2]
  3. Find P(\(W\) | \(F\)) and explain what this probability represents. [3]
OCR MEI S1 2011 June Q6
7 marks Moderate -0.8
The numbers of eggs laid by a sample of 70 female herring gulls are shown in the table.
Number of eggs1234
Frequency1040155
  1. Find the mean and standard deviation of the number of eggs laid per gull. [5]
  2. The sample did not include female herring gulls that laid no eggs. How would the mean and standard deviation change if these gulls were included? [2]
OCR MEI S1 2011 June Q7
18 marks Standard +0.3
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
    1. there is exactly 1 no-show in the sample, [3]
    2. there are at least 2 no-shows in the sample. [2]
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.
  1. Write down suitable null and alternative hypotheses for the test. Give a reason for your choice of alternative hypothesis. [4]
  2. In the case that \(n = 20\) and the number of no-shows in the sample is 1, carry out the test. [4]
  3. 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. [3]
  4. In the case that \(n \leqslant 18\), explain why there is no point in carrying out the test at the 5% level. [2]
OCR MEI S1 2011 June Q8
18 marks Moderate -0.3
The heating quality of the coal in a sample of 50 sacks is measured in suitable units. The data are summarised below.
Heating quality (\(x\))9.1 \(\leqslant x <\) 9.39.3 \(< x \leqslant\) 9.59.5 \(< x \leqslant\) 9.79.7 \(< x \leqslant\) 9.99.9 \(< x \leqslant\) 10.1
Frequency5715167
  1. Draw a cumulative frequency diagram to illustrate these data. [5]
  2. Use the diagram to estimate the median and interquartile range of the data. [3]
  3. Show that there are no outliers in the sample. [3]
  4. Three of these 50 sacks are selected at random. Find the probability that
    1. in all three, the heating quality \(x\) is more than 9.5, [3]
    2. in at least two, the heating quality \(x\) is more than 9.5. [4]
OCR MEI S1 2014 June Q1
8 marks Easy -1.3
The ages, \(x\) years, of the senior members of a running club are summarised in the table below.
Age (\(x\))\(20 \leqslant x < 30\)\(30 \leqslant x < 40\)\(40 \leqslant x < 50\)\(50 \leqslant x < 60\)\(60 \leqslant x < 70\)\(70 \leqslant x < 80\)\(80 \leqslant x < 90\)
Frequency10304223951
  1. Draw a cumulative frequency diagram to illustrate the data. [5]
  2. Use your diagram to estimate the median and interquartile range of the data. [3]
OCR MEI S1 2014 June Q2
8 marks Moderate -0.8
Candidates applying for jobs in a large company take an aptitude test, as a result of which they are either accepted, rejected or retested, with probabilities 0.2, 0.5 and 0.3 respectively. When a candidate is retested for the first time, the three possible outcomes and their probabilities remain the same as for the original test. When a candidate is retested for the second time there are just two possible outcomes, accepted or rejected, with probabilities 0.4 and 0.6 respectively.
  1. Draw a probability tree diagram to illustrate the outcomes. [3]
  2. Find the probability that a randomly selected candidate is accepted. [2]
  3. Find the probability that a randomly selected candidate is retested at least once, given that this candidate is accepted. [3]
OCR MEI S1 2014 June Q3
6 marks Easy -1.2
Each weekday, Marta travels to school by bus. Sometimes she arrives late. • \(L\) is the event that Marta arrives late. • \(R\) is the event that it is raining. You are given that \(\mathrm{P}(L) = 0.15\), \(\mathrm{P}(R) = 0.22\) and \(\mathrm{P}(L \mid R) = 0.45\).
  1. Use this information to show that the events \(L\) and \(R\) are not independent. [1]
  2. Find \(\mathrm{P}(L \cap R)\). [2]
  3. Draw a Venn diagram showing the events \(L\) and \(R\), and fill in the probability corresponding to each of the four regions of your diagram. [3]
OCR MEI S1 2014 June Q4
6 marks Moderate -0.8
There are 16 girls and 14 boys in a class. Four of them are to be selected to form a quiz team. The team is to be selected at random.
  1. Find the probability that all 4 members of the team will be girls. [3]
  2. Find the probability that the team will contain at least one girl and at least one boy. [3]
OCR MEI S1 2014 June Q5
8 marks Moderate -0.8
The probability distribution of the random variable \(X\) is given by the formula $$\mathrm{P}(X = r) = k + 0.01r^2 \text{ for } r = 1, 2, 3, 4, 5.$$
  1. Show that \(k = 0.09\). Using this value of \(k\), display the probability distribution of \(X\) in a table. [3]
  2. Find \(\mathrm{E}(X)\) and \(\mathrm{Var}(X)\). [5]
OCR MEI S1 2014 June Q6
17 marks Moderate -0.8
The weights, \(w\) grams, of a random sample of 60 carrots of variety A are summarised in the table below.
Weight\(30 \leqslant w < 50\)\(50 \leqslant w < 60\)\(60 \leqslant w < 70\)\(70 \leqslant w < 80\)\(80 \leqslant w < 90\)
Frequency111018147
  1. Draw a histogram to illustrate these data. [5]
  2. Calculate estimates of the mean and standard deviation of \(w\). [4]
  3. Use your answers to part (ii) to investigate whether there are any outliers. [3]
The weights, \(x\) grams, of a random sample of 50 carrots of variety B are summarised as follows. $$n = 50 \quad \sum x = 3624.5 \quad \sum x^2 = 265416$$
  1. Calculate the mean and standard deviation of \(x\). [3]
  2. Compare the central tendency and variation of the weights of varieties A and B. [2]
OCR MEI S1 2014 June Q7
19 marks Standard +0.3
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. Find the probability that exactly 12 germinate. [3]
    2. Find the probability that fewer than 12 germinate. [2]
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.
  1. Write down suitable null and alternative hypotheses for the test. Give a reason for your choice of alternative hypothesis. [4]
  2. In a trial with \(n = 20\), Ramesh finds that 13 seeds germinate. Carry out the test. [4]
  3. 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. [3]
  4. 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. [3]
Edexcel S1 Q1
8 marks Moderate -0.8
An athlete believes that her times for running 200 metres in races are normally distributed with a mean of 22.8 seconds.
  1. Given that her time is over 23.3 seconds in 20\% of her races, calculate the variance of her times. [5]
  2. The record over this distance for women at her club is 21.82 seconds. According to her model, what is the chance that she will beat this record in her next race? [3]
Edexcel S1 Q2
10 marks Moderate -0.3
The events \(A\) and \(B\) are such that $$\text{P}(A) = \frac{5}{16}, \text{P}(B) = \frac{1}{2} \text{ and P}(A|B) = \frac{1}{4}$$ Find
  1. P\((A \cap B)\). [2]
  2. P\((B'|A)\). [3]
  3. P\((A' \cup B)\). [2]
  4. Determine, with a reason, whether or not the events \(A\) and \(B\) are independent. [3]
Edexcel S1 Q3
10 marks Moderate -0.3
A group of 60 children were each asked to choose an integer value between 1 and 9 inclusive. Their choices are summarised in the table below.
Value chosen123456789
Number of children345101213742
  1. Calculate the mean and standard deviation of the values chosen. [6]
It is suggested that the value chosen could be modelled by a discrete uniform distribution.
  1. Write down the mean that this model would predict. [2]
Given also that the standard deviation according to this model would be 2.58,
  1. explain why this model is not suitable and suggest why this is the case. [2]
Edexcel S1 Q4
13 marks Moderate -0.3
A six-sided die is biased such that there is an equal chance of scoring each of the numbers from 1 to 5 but a score of 6 is three times more likely than each of the other numbers.
  1. Write down the probability distribution for the random variable, \(X\), the score on a single throw of the die. [4]
  2. Show that E\((X) = \frac{33}{8}\). [3]
  3. Find E\((4X - 1)\). [2]
  4. Find Var\((X)\). [4]
Edexcel S1 Q5
17 marks Moderate -0.3
The number of patients attending a hospital trauma clinic each day was recorded over several months, giving the data in the table below.
Number of patients10 - 1920 - 2930 - 3435 - 3940 - 4445 - 4950 - 69
Frequency218243027145
These data are represented by a histogram. Given that the bar representing the 20 - 29 group is 2 cm wide and 7.2 cm high,
  1. calculate the dimensions of the bars representing the groups
    1. 30 - 34
    2. 50 - 69
    [6]
  2. Use linear interpolation to estimate the median and quartiles of these data. [6]
The lowest and highest numbers of patients recorded were 14 and 67 respectively.
  1. Represent these data with a boxplot drawn on graph paper and describe the skewness of the distribution. [5]
Edexcel S1 Q6
17 marks Moderate -0.3
Penshop have stores selling stationary in each of 6 towns. The population, \(P\), in tens of thousands and the monthly turnover, \(T\), in thousands of pounds for each of the shops are as recorded below.
TownAbbertonBemberClasterDellerEdgetonFigland
\(P\) (0.000's)3.27.65.29.08.14.8
\(T\) (£ 000's)11.112.413.319.317.911.8
  1. Represent these data on a scatter diagram with \(T\) on the vertical axis. [4]
    1. Which town's shop might appear to be underachieving given the populations of the towns?
    2. Suggest two other factors that might affect each shop's turnover. [3]
You may assume that $$\Sigma P = 37.9, \quad \Sigma T = 85.8, \quad \Sigma P^2 = 264.69, \quad \Sigma T^2 = 1286, \quad \Sigma PT = 574.25.$$
  1. Find the equation of the regression line of \(T\) on \(P\). [7]
  2. Estimate the monthly turnover that might be expected if a shop were opened in Gratton, a town with a population of 68 000. [2]
  3. Why might the management of Penshop be reluctant to use the regression line to estimate the monthly turnover they could expect if a shop were opened in Haggin, a town with a population of 172 000? [1]
Edexcel S1 Q1
6 marks Moderate -0.8
  1. Draw two separate scatter diagrams, each with eight points, to illustrate the relationship between \(x\) and \(y\) in the cases where they have a product moment correlation coefficient equal to
    1. exactly \(+1\),
    2. about \(-0.4\). [4 marks]
  2. Explain briefly how the conclusion you would draw from a product moment correlation coefficient of \(+0.3\) would vary according to the number of pairs of data used in its calculation. [2 marks]
Edexcel S1 Q2
6 marks Moderate -0.8
A histogram was drawn to show the distribution of age in completed years of the participants on an outward-bound course. There were 32 people aged 30-34 years on the course. The height of the rectangle representing this group was 19.2 cm and it was 1 cm in width. Given that there were 28 people aged 35-39 years,
  1. find the height of the rectangle representing this group. [3 marks]
Given that the height of the rectangle representing people aged 40-59 years was 2.7 cm,
  1. find the number of people on the course in this age group. [3 marks]
Edexcel S1 Q3
9 marks Moderate -0.3
The events \(A\) and \(B\) are such that $$\text{P}(A) = \frac{7}{12}, \quad \text{P}(A \cap B) = \frac{1}{4} \quad \text{and} \quad \text{P}(A|B) = \frac{2}{3}.$$ Find
  1. P\((B)\), [3 marks]
  2. P\((A \cup B)\), [3 marks]
  3. P\((B|A')\). [3 marks]
Edexcel S1 Q4
12 marks Standard +0.3
The owner of a mobile burger-bar believes that hot weather reduces his sales. To investigate the effect on his business he collected data on his daily sales, \(£P\), and the maximum temperature, \(T\)°C, on each of 20 days. He then coded the data, using \(x = T - 20\) and \(y = P - 300\), and calculated the summary statistics given below. $$\Sigma x = 57, \quad \Sigma y = 2222, \quad \Sigma x^2 = 401, \quad \Sigma y^2 = 305576, \quad \Sigma xy = 3871.$$
  1. Find an equation of the regression line of \(P\) on \(T\). [9 marks]
The owner of the bar doesn't believe it is profitable for him to run the bar if he takes less than £460 in a day.
  1. According to your regression line at what maximum daily temperature, to the nearest degree Celsius, does it become unprofitable for him to run the bar? [3 marks]
Edexcel S1 Q5
13 marks Moderate -0.8
The discrete random variable \(X\) has the probability function shown below. $$P(X = x) = \begin{cases} kx, & x = 2, 3, 4, 5, 6, \\ 0, & \text{otherwise}. \end{cases}$$
  1. Find the value of \(k\). [2 marks]
  2. Show that E\((X) = \frac{9}{2}\). [3 marks]
Find
  1. P\([X > \text{E}(X)]\), [2 marks]
  2. E\((2X - 5)\), [2 marks]
  3. Var\((X)\). [4 marks]
Edexcel S1 Q6
14 marks Standard +0.3
A geologist is analysing the size of quartz crystals in a sample of granite. She estimates that the longest diameter of 75% of the crystals is greater than 2 mm, but only 10% of the crystals have a longest diameter of more than 6 mm. The geologist believes that the distribution of the longest diameters of the quartz crystals can be modelled by a normal distribution.
  1. Find the mean and variance of this normal distribution. [9 marks]
The geologist also estimated that only 2% of the longest diameters were smaller than 1 mm.
  1. Calculate the corresponding percentage that would be predicted by a normal distribution with the parameters you calculated in part \((a)\). [3 marks]
  2. Hence, comment on the suitability of the normal distribution as a model in this situation. [2 marks]
Edexcel S1 Q7
15 marks Moderate -0.8
Jane and Tahira play together in a basketball team. The list below shows the number of points that Jane scored in each of 30 games.
39192830182123153424
29174312242541192640
45232132372418152436
  1. Construct a stem and leaf diagram for these data. [3 marks]
  2. Find the median and quartiles for these data. [4 marks]
  3. Represent these data with a boxplot. [3 marks]
Tahira played in the same 30 games and her lowest and highest points total in a game were 19 and 41 respectively. The quartiles for Tahira were 27, 31 and 35 respectively.
  1. Using the same scale draw a boxplot for Tahira's points totals. [2 marks]
  2. Compare and contrast the number of points scored per game by Jane and Tahira. [3 marks]