Questions — Edexcel (9685 questions)

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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 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 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 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 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
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Edexcel S2 2009 January Q5
9 marks Standard +0.3
  1. A factory produces components of which \(1 \%\) are defective. The components are packed in boxes of 10 . A box is selected at random.
    1. Find the probability that the box contains exactly one defective component.
    2. Find the probability that there are at least 2 defective components in the box.
    3. Using a suitable approximation, find the probability that a batch of 250 components contains between 1 and 4 (inclusive) defective components.
    4. A web server is visited on weekdays, at a rate of 7 visits per minute. In a random one minute on a Saturday the web server is visited 10 times.
      1. Test, at the \(10 \%\) level of significance, whether or not there is evidence that the rate of visits is greater on a Saturday than on weekdays. State your hypotheses clearly.
      2. State the minimum number of visits required to obtain a significant result.
    6. State an assumption that has been made about the visits to the server.
    In a random two minute period on a Saturday the web server is visited 20 times.
  2. Using a suitable approximation, test at the \(10 \%\) level of significance, whether or not the rate of visits is greater on a Saturday.
Edexcel S2 2009 January Q7
13 marks Moderate -0.3
  1. A random variable \(X\) has probability density function given by
$$f ( x ) = \left\{ \begin{array} { c l } - \frac { 2 } { 9 } x + \frac { 8 } { 9 } & 1 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{array} \right.$$
  1. Show that the cumulative distribution function \(\mathrm { F } ( x )\) can be written in the form \(a x ^ { 2 } + b x + c\), for \(1 \leqslant x \leqslant 4\) where \(a , b\) and \(c\) are constants.
  2. Define fully the cumulative distribution function \(\mathrm { F } ( x )\).
  3. Show that the upper quartile of \(X\) is 2.5 and find the lower quartile. Given that the median of \(X\) is 1.88
  4. describe the skewness of the distribution. Give a reason for your answer.
Edexcel S2 2010 January Q1
8 marks Easy -1.2
  1. A manufacturer supplies DVD players to retailers in batches of 20 . It has \(5 \%\) of the players returned because they are faulty.
    1. Write down a suitable model for the distribution of the number of faulty DVD players in a batch.
    Find the probability that a batch contains
  2. no faulty DVD players,
  3. more than 4 faulty DVD players.
  4. Find the mean and variance of the number of faulty DVD players in a batch.
Edexcel S2 2010 January Q2
10 marks Moderate -0.8
2. A continuous random variable \(X\) has cumulative distribution function $$\mathrm { F } ( x ) = \begin{cases} 0 , & x < - 2 \\ \frac { x + 2 } { 6 } , & - 2 \leqslant x \leqslant 4 \\ 1 , & x > 4 \end{cases}$$
  1. Find \(\mathrm { P } ( X < 0 )\).
  2. Find the probability density function \(\mathrm { f } ( x )\) of \(X\).
  3. Write down the name of the distribution of \(X\).
  4. Find the mean and the variance of \(X\).
  5. Write down the value of \(\mathrm { P } ( X = 1 )\).
Edexcel S2 2010 January Q3
10 marks Moderate -0.3
  1. A robot is programmed to build cars on a production line. The robot breaks down at random at a rate of once every 20 hours.
    1. Find the probability that it will work continuously for 5 hours without a breakdown.
    Find the probability that, in an 8 hour period,
  2. the robot will break down at least once,
  3. there are exactly 2 breakdowns. In a particular 8 hour period, the robot broke down twice.
  4. Write down the probability that the robot will break down in the following 8 hour period. Give a reason for your answer.
Edexcel S2 2010 January Q4
17 marks Standard +0.3
  1. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by
$$f ( x ) = \begin{cases} k \left( x ^ { 2 } - 2 x + 2 \right) & 0 < x \leqslant 3 \\ 3 k & 3 < x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 1 } { 9 }\)
  2. Find the cumulative distribution function \(\mathrm { F } ( x )\).
  3. Find the mean of \(X\).
  4. Show that the median of \(X\) lies between \(x = 2.6\) and \(x = 2.7\)
Edexcel S2 2010 January Q5
9 marks Standard +0.3
  1. A café serves breakfast every morning. Customers arrive for breakfast at random at a rate of 1 every 6 minutes.
Find the probability that
  1. fewer than 9 customers arrive for breakfast on a Monday morning between 10 am and 11 am. The café serves breakfast every day between 8 am and 12 noon.
  2. Using a suitable approximation, estimate the probability that more than 50 customers arrive for breakfast next Tuesday.
Edexcel S2 2010 January Q6
10 marks Standard +0.3
6. (a) Define the critical region of a test statistic. A discrete random variable \(X\) has a Binomial distribution \(\mathrm { B } ( 30 , p )\). A single observation is used to test \(\mathrm { H } _ { 0 } : p = 0.3\) against \(\mathrm { H } _ { 1 } : p \neq 0.3\) (b) Using a \(1 \%\) level of significance find the critical region of this test. You should state the probability of rejection in each tail which should be as close as possible to 0.005
(c) Write down the actual significance level of the test. The value of the observation was found to be 15 .
(d) Comment on this finding in light of your critical region.
Edexcel S2 2010 January Q7
11 marks Moderate -0.5
  1. A bag contains a large number of coins. It contains only \(1 p\) and \(2 p\) coins in the ratio \(1 : 3\)
    1. Find the mean \(\mu\) and the variance \(\sigma ^ { 2 }\) of the values of this population of coins.
    A random sample of size 3 is taken from the bag.
  2. List all the possible samples.
  3. Find the sampling distribution of the mean value of the samples.
Edexcel S2 2011 January Q1
10 marks Moderate -0.3
  1. A disease occurs in \(3 \%\) of a population.
    1. State any assumptions that are required to model the number of people with the disease in a random sample of size \(n\) as a binomial distribution.
    2. Using this model, find the probability of exactly 2 people having the disease in a random sample of 10 people.
    3. Find the mean and variance of the number of people with the disease in a random sample of 100 people.
    A doctor tests a random sample of 100 patients for the disease. He decides to offer all patients a vaccination to protect them from the disease if more than 5 of the sample have the disease.
  2. Using a suitable approximation, find the probability that the doctor will offer all patients a vaccination.
Edexcel S2 2011 January Q2
6 marks Moderate -0.8
2. A student takes a multiple choice test. The test is made up of 10 questions each with 5 possible answers. The student gets 4 questions correct. Her teacher claims she was guessing the answers. Using a one tailed test, at the \(5 \%\) level of significance, test whether or not there is evidence to reject the teacher's claim. State your hypotheses clearly.
Edexcel S2 2011 January Q3
11 marks Standard +0.3
3. The continuous random variable \(X\) is uniformly distributed over the interval \([ - 1,3 ]\). Find
  1. \(\mathrm { E } ( X )\)
  2. \(\operatorname { Var } ( X )\)
  3. \(\mathrm { E } \left( X ^ { 2 } \right)\)
  4. \(\mathrm { P } ( X < 1.4 )\) A total of 40 observations of \(X\) are made.
  5. Find the probability that at least 10 of these observations are negative.
Edexcel S2 2011 January Q4
6 marks Moderate -0.3
  1. Richard regularly travels to work on a ferry. Over a long period of time, Richard has found that the ferry is late on average 2 times every week. The company buys a new ferry to improve the service. In the 4-week period after the new ferry is launched, Richard finds the ferry is late 3 times and claims the service has improved. Assuming that the number of times the ferry is late has a Poisson distribution, test Richard's claim at the \(5 \%\) level of significance. State your hypotheses clearly.
    (6)
  2. A continuous random variable \(X\) has the probability density function \(\mathrm { f } ( x )\) shown in Figure 1.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{58e5aa9e-f177-48ad-8bb8-54c0e2c21e6d-07_591_689_358_630} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure}
  1. Show that \(\mathrm { f } ( x ) = 4 - 8 x\) for \(0 \leqslant x \leqslant 0.5\) and specify \(\mathrm { f } ( x )\) for all real values of \(x\).
  2. Find the cumulative distribution function \(\mathrm { F } ( x )\).
  3. Find the median of \(X\).
  4. Write down the mode of \(X\).
  5. State, with a reason, the skewness of \(X\).
Edexcel S2 2011 January Q6
16 marks Standard +0.3
6. Cars arrive at a motorway toll booth at an average rate of 150 per hour.
  1. Suggest a suitable distribution to model the number of cars arriving at the toll booth, \(X\), per minute.
  2. State clearly any assumptions you have made by suggesting this model. Using your model,
  3. find the probability that in any given minute
    1. no cars arrive,
    2. more than 3 cars arrive.
  4. In any given 4 minute period, find \(m\) such that \(\mathrm { P } ( X > m ) = 0.0487\)
  5. Using a suitable approximation find the probability that fewer than 15 cars arrive in any given 10 minute period.
    January 2011
Edexcel S2 2011 January Q7
13 marks Standard +0.3
7. The queuing time in minutes, \(X\), of a customer at a post office is modelled by the probability density function $$f ( x ) = \begin{cases} k x \left( 81 - x ^ { 2 } \right) & 0 \leqslant x \leqslant 9 \\ 0 & \text { otherwise } \end{cases}$$
  1. Show that \(k = \frac { 4 } { 6561 }\). Using integration, find
  2. the mean queuing time of a customer,
  3. the probability that a customer will queue for more than 5 minutes. Three independent customers shop at the post office.
  4. Find the probability that at least 2 of the customers queue for more than 5 minutes.
Edexcel S2 2012 January Q1
8 marks Moderate -0.8
  1. The time in minutes that Elaine takes to checkout at her local supermarket follows a continuous uniform distribution defined over the interval [3,9].
Find
  1. Elaine's expected checkout time,
  2. the variance of the time taken to checkout at the supermarket,
  3. the probability that Elaine will take more than 7 minutes to checkout. Given that Elaine has already spent 4 minutes at the checkout,
  4. find the probability that she will take a total of less than 6 minutes to checkout.
Edexcel S2 2012 January Q2
7 marks Moderate -0.3
2. David claims that the weather forecasts produced by local radio are no better than those achieved by tossing a fair coin and predicting rain if a head is obtained or no rain if a tail is obtained. He records the weather for 30 randomly selected days. The local radio forecast is correct on 21 of these days. Test David's claim at the \(5 \%\) level of significance. State your hypotheses clearly.
Edexcel S2 2012 January Q3
9 marks Moderate -0.8
3. The probability of a telesales representative making a sale on a customer call is 0.15 Find the probability that
  1. no sales are made in 10 calls,
  2. more than 3 sales are made in 20 calls. Representatives are required to achieve a mean of at least 5 sales each day.
  3. Find the least number of calls each day a representative should make to achieve this requirement.
  4. Calculate the least number of calls that need to be made by a representative for the probability of at least 1 sale to exceed 0.95
Edexcel S2 2012 January Q4
16 marks Moderate -0.3
4. A website receives hits at a rate of 300 per hour.
  1. State a distribution that is suitable to model the number of hits obtained during a 1 minute interval.
  2. State two reasons for your answer to part (a). Find the probability of
  3. 10 hits in a given minute,
  4. at least 15 hits in 2 minutes. The website will go down if there are more than 70 hits in 10 minutes.
  5. Using a suitable approximation, find the probability that the website will go down in a particular 10 minute interval.
Edexcel S2 2012 January Q5
7 marks Moderate -0.3
  1. The probability of an electrical component being defective is 0.075 The component is supplied in boxes of 120
    1. Using a suitable approximation, estimate the probability that there are more than 3 defective components in a box.
    A retailer buys 2 boxes of components.
  2. Estimate the probability that there are at least 4 defective components in each box.
Edexcel S2 2012 January Q6
18 marks Standard +0.3
6. A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { 2 } & 0 \leqslant x < 1 \\ x - \frac { 1 } { 2 } & 1 \leqslant x \leqslant k \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a positive constant.
  1. Sketch the graph of \(\mathrm { f } ( x )\).
  2. Show that \(k = \frac { 1 } { 2 } ( 1 + \sqrt { 5 } )\).
  3. Define fully the cumulative distribution function \(\mathrm { F } ( x )\).
  4. Find \(\mathrm { P } ( 0.5 < X < 1.5 )\).
  5. Write down the median of \(X\) and the mode of \(X\).
  6. Describe the skewness of the distribution of \(X\). Give a reason for your answer.
Edexcel S2 2012 January Q7
10 marks Standard +0.3
7. (a) Explain briefly what you understand by
  1. a critical region of a test statistic,
  2. the level of significance of a hypothesis test.
    (b) An estate agent has been selling houses at a rate of 8 per month. She believes that the rate of sales will decrease in the next month.
  3. Using a \(5 \%\) level of significance, find the critical region for a one tailed test of the hypothesis that the rate of sales will decrease from 8 per month.
  4. Write down the actual significance level of the test in part (b)(i). The estate agent is surprised to find that she actually sold 13 houses in the next month. She now claims that this is evidence of an increase in the rate of sales per month.
    (c) Test the estate agent's claim at the \(5 \%\) level of significance. State your hypotheses clearly.
Edexcel S2 2013 January Q1
5 marks Easy -1.2
  1. (a) Write down the conditions under which the Poisson distribution can be used as an approximation to the binomial distribution.
The probability of any one letter being delivered to the wrong house is 0.01 On a randomly selected day Peter delivers 1000 letters.
(b) Using a Poisson approximation, find the probability that Peter delivers at least 4 letters to the wrong house. Give your answer to 4 decimal places.
Edexcel S2 2013 January Q2
11 marks Moderate -0.3
2. In a village, power cuts occur randomly at a rate of 3 per year.
  1. Find the probability that in any given year there will be
    1. exactly 7 power cuts,
    2. at least 4 power cuts.
  2. Use a suitable approximation to find the probability that in the next 10 years the number of power cuts will be less than 20
Edexcel S2 2013 January Q3
10 marks Standard +0.3
  1. A random variable \(X\) has the distribution \(\mathrm { B } ( 12 , p )\).
    1. Given that \(p = 0.25\) find
      1. \(\mathrm { P } ( X < 5 )\)
      2. \(\mathrm { P } ( X \geqslant 7 )\)
    2. Given that \(\mathrm { P } ( X = 0 ) = 0.05\), find the value of \(p\) to 3 decimal places.
    3. Given that the variance of \(X\) is 1.92 , find the possible values of \(p\).