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 2004 June Q6
14 marks Moderate -0.3
6. Minor defects occur in a particular make of carpet at a mean rate of 0.05 per \(\mathrm { m } ^ { 2 }\).
  1. Suggest a suitable model for the distribution of the number of defects in this make of carpet. Give a reason for your answer. A carpet fitter has a contract to fit this carpet in a small hotel. The hotel foyer requires \(30 \mathrm {~m} ^ { 2 }\) of this carpet. Find the probability that the foyer carpet contains
  2. exactly 2 defects,
  3. more than 5 defects. The carpet fitter orders a total of \(355 \mathrm {~m} ^ { 2 }\) of the carpet for the whole hotel.
  4. Using a suitable approximation, find the probability that this total area of carpet contains 22 or more defects.
    (6)
Edexcel S2 2004 June Q7
17 marks Standard +0.3
7. A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { 3 } , & 0 \leq x \leq 1 \\ \frac { 8 x ^ { 3 } } { 45 } , & 1 \leq x \leq 2 \\ 0 , & \text { otherwise } \end{cases}$$
  1. Calculate the mean of \(X\).
  2. Specify fully the cumulative distribution function \(\mathrm { F } ( x )\).
  3. Find the median of \(X\).
  4. Comment on the skewness of the distribution of \(X\). END
Edexcel S2 2005 June Q1
7 marks Easy -1.2
  1. It is estimated that \(4 \%\) of people have green eyes. In a random sample of size \(n\), the expected number of people with green eyes is 5 .
    1. Calculate the value of \(n\).
    The expected number of people with green eyes in a second random sample is 3 .
  2. Find the standard deviation of the number of people with green eyes in this second sample. expected number of people with green eyes is 5 .
  3. Calculate the value of \(n\) - The expected number of people with green eyes in a second random sample is 3 .
  4. sample. C) T. " D
Edexcel S2 2005 June Q2
11 marks Easy -1.3
2. The continuous random variable \(X\) is uniformly distributed over the interval \([ 2,6 ]\).
  1. Write down the probability density function \(\mathrm { f } ( x )\). Find
  2. \(\mathrm { E } ( X )\),
  3. \(\operatorname { Var } ( X )\),
  4. the cumulative distribution function of \(X\), for all \(x\),
  5. \(\mathrm { P } ( 2.3 < X < 3.4 )\).
Edexcel S2 2005 June Q3
14 marks Easy -1.2
3. The random variable \(X\) is the number of misprints per page in the first draft of a novel.
  1. State two conditions under which a Poisson distribution is a suitable model for \(X\). The number of misprints per page has a Poisson distribution with mean 2.5. Find the probability that
  2. a randomly chosen page has no misprints,
  3. the total number of misprints on 2 randomly chosen pages is more than 7 . The first chapter contains 20 pages.
  4. Using a suitable approximation find, to 2 decimal places, the probability that the chapter will contain less than 40 misprints.
Edexcel S2 2005 June Q4
4 marks Easy -1.8
4. Explain what you understand by
  1. a sampling unit,
  2. a sampling frame,
  3. a sampling distribution.
Edexcel S2 2005 June Q5
7 marks Moderate -0.8
5. In a manufacturing process, \(2 \%\) of the articles produced are defective. A batch of 200 articles is selected.
  1. Giving a justification for your choice, use a suitable approximation to estimate the probability that there are exactly 5 defective articles.
  2. Estimate the probability there are less than 5 defective articles.
Edexcel S2 2005 June Q6
18 marks Standard +0.3
6. A continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) where $$f ( x ) = \begin{cases} k \left( 4 x - x ^ { 3 } \right) , & 0 \leqslant x \leqslant 2 \\ 0 , & \text { otherwise } \end{cases}$$ where \(k\) is a positive integer.
  1. Show that \(k = \frac { 1 } { 4 }\). Find
  2. \(\mathrm { E } ( X )\),
  3. the mode of \(X\),
  4. the median of \(X\).
  5. Comment on the skewness of the distribution.
  6. Sketch f(x).
Edexcel S2 2005 June Q7
14 marks Standard +0.3
7. A drugs company claims that \(75 \%\) of patients suffering from depression recover when treated with a new drug. A random sample of 10 patients with depression is taken from a doctor's records.
  1. Write down a suitable distribution to model the number of patients in this sample who recover when treated with the new drug. Given that the claim is correct,
  2. find the probability that the treatment will be successful for exactly 6 patients. The doctor believes that the claim is incorrect and the percentage who will recover is lower. From her records she took a random sample of 20 patients who had been treated with the new drug. She found that 13 had recovered.
  3. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, the doctor's belief.
  4. From a sample of size 20, find the greatest number of patients who need to recover for the test in part (c) to be significant at the \(1 \%\) level.
Edexcel S2 2006 June Q1
3 marks Moderate -0.8
  1. Before introducing a new rule the secretary of a golf club decided to find out how members might react to this rule.
    1. Explain why the secretary decided to take a random sample of club members rather than ask all the members.
    2. Suggest a suitable sampling frame.
    3. Identify the sampling units. \includegraphics[max width=\textwidth, alt={}, center]{992812ed-58bb-4f47-ad51-f748c8312336-02_102_1831_2650_114}
    4. The continuous random variable \(L\) represents the error, in mm , made when a machine cuts rods to a target length. The distribution of \(L\) is continuous uniform over the interval [-4.0, 4.0].
    Find
  2. \(\mathrm { P } ( L < - 2.6 )\),
  3. \(\mathrm { P } ( L < - 3.0\) or \(L > 3.0 )\). A random sample of 20 rods cut by the machine was checked.
  4. Find the probability that more than half of them were within 3.0 mm of the target length.
Edexcel S2 2006 June Q3
11 marks Standard +0.3
3. An estate agent sells properties at a mean rate of 7 per week.
  1. Suggest a suitable model to represent the number of properties sold in a randomly chosen week. Give two reasons to support your model.
  2. Find the probability that in any randomly chosen week the estate agent sells exactly 5 properties.
  3. Using a suitable approximation find the probability that during a 24 week period the estate agent sells more than 181 properties.
Edexcel S2 2006 June Q4
11 marks Standard +0.3
  1. Breakdowns occur on a particular machine at random at a mean rate of 1.25 per week.
    1. Find the probability that fewer than 3 breakdowns occurred in a randomly chosen week.
    Over a 4 week period the machine was monitored. During this time there were 11 breakdowns.
  2. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the rate of breakdowns has changed over this period. State your hypotheses clearly.
Edexcel S2 2006 June Q5
13 marks Moderate -0.3
  1. A manufacturer produces large quantities of coloured mugs. It is known from previous records that \(6 \%\) of the production will be green.
A random sample of 10 mugs was taken from the production line.
  1. Define a suitable distribution to model the number of green mugs in this sample.
  2. Find the probability that there were exactly 3 green mugs in the sample. A random sample of 125 mugs was taken.
  3. Find the probability that there were between 10 and 13 (inclusive) green mugs in this sample, using
    1. a Poisson approximation,
    2. a Normal approximation.
Edexcel S2 2006 June Q6
16 marks Standard +0.3
6. The continuous random variable \(X\) has probability density function $$f ( x ) = \left\{ \begin{array} { c c } \frac { 1 + x } { k } , & 1 \leqslant x \leqslant 4 \\ 0 , & \text { otherwise } \end{array} \right.$$
  1. Show that \(k = \frac { 21 } { 2 }\).
  2. Specify fully the cumulative distribution function of \(X\).
  3. Calculate \(\mathrm { E } ( X )\).
  4. Find the value of the median.
  5. Write down the mode.
  6. Explain why the distribution is negatively skewed.
Edexcel S2 2006 June Q7
14 marks Standard +0.3
  1. It is known from past records that 1 in 5 bowls produced in a pottery have minor defects. To monitor production a random sample of 25 bowls was taken and the number of such bowls with defects was recorded.
    1. Using a 5\% level of significance, find critical regions for a two-tailed test of the hypothesis that 1 in 5 bowls have defects. The probability of rejecting, in either tail, should be as close to \(2.5 \%\) as possible.
    2. State the actual significance level of the above test.
    At a later date, a random sample of 20 bowls was taken and 2 of them were found to have defects.
  2. Test, at the \(10 \%\) level of significance, whether or not there is evidence that the proportion of bowls with defects has decreased. State your hypotheses clearly.
Edexcel S2 2009 June Q1
5 marks Standard +0.3
  1. A bag contains a large number of counters of which \(15 \%\) are coloured red. A random sample of 30 counters is selected and the number of red counters is recorded.
    1. Find the probability of no more than 6 red counters in this sample.
    A second random sample of 30 counters is selected and the number of red counters is recorded.
  2. Using a Poisson approximation, estimate the probability that the total number of red counters in the combined sample of size 60 is less than 13.
Edexcel S2 2009 June Q2
6 marks Standard +0.3
2. An effect of a certain disease is that a small number of the red blood cells are deformed. Emily has this disease and the deformed blood cells occur randomly at a rate of 2.5 per ml of her blood. Following a course of treatment, a random sample of 2 ml of Emily's blood is found to contain only 1 deformed red blood cell. Stating your hypotheses clearly and using a \(5 \%\) level of significance, test whether or not there has been a decrease in the number of deformed red blood cells in Emily's blood.
Edexcel S2 2009 June Q3
5 marks Moderate -0.8
3. A random sample \(X _ { 1 } , X _ { 2 } , \ldots X _ { n }\) is taken from a population with unknown mean \(\mu\) and unknown variance \(\sigma ^ { 2 }\). A statistic \(Y\) is based on this sample.
  1. Explain what you understand by the statistic \(Y\).
  2. Explain what you understand by the sampling distribution of \(Y\).
  3. State, giving a reason which of the following is not a statistic based on this sample.
    1. \(\sum _ { i = 1 } ^ { n } \frac { \left( X _ { i } - \bar { X } \right) ^ { 2 } } { n }\)
    2. \(\sum _ { i = 1 } ^ { n } \left( \frac { X _ { i } - \mu } { \sigma } \right) ^ { 2 }\)
    3. \(\sum _ { i = 1 } ^ { n } X _ { i } ^ { 2 }\)
Edexcel S2 2009 June Q4
8 marks Standard +0.3
4. Past records suggest that \(30 \%\) of customers who buy baked beans from a large supermarket buy them in single tins. A new manager questions whether or not there has been a change in the proportion of customers who buy baked beans in single tins. A random sample of 20 customers who had bought baked beans was taken.
  1. Using a \(10 \%\) level of significance, find the critical region for a two-tailed test to answer the manager's question. You should state the probability of rejection in each tail which should be less than 0.05 .
  2. Write down the actual significance level of a test based on your critical region from part (a). The manager found that 11 customers from the sample of 20 had bought baked beans in single tins.
  3. Comment on this finding in the light of your critical region found in part (a).
Edexcel S2 2009 June Q5
10 marks Standard +0.3
  1. An administrator makes errors in her typing randomly at a rate of 3 errors every 1000 words.
    1. In a document of 2000 words find the probability that the administrator makes 4 or more errors.
    The administrator is given an 8000 word report to type and she is told that the report will only be accepted if there are 20 or fewer errors.
  2. Use a suitable approximation to calculate the probability that the report is accepted.
Edexcel S2 2009 June Q6
13 marks Standard +0.3
6. The three independent random variables \(A , B\) and \(C\) each has a continuous uniform distribution over the interval \([ 0,5 ]\).
  1. Find \(\mathrm { P } ( A > 3 )\).
  2. Find the probability that \(A , B\) and \(C\) are all greater than 3 . The random variable \(Y\) represents the maximum value of \(A , B\) and \(C\). The cumulative distribution function of \(Y\) is $$\mathrm { F } ( y ) = \begin{cases} 0 & y < 0 \\ \frac { y ^ { 3 } } { 125 } & 0 \leqslant y \leqslant 5 \\ 1 & y > 5 \end{cases}$$
  3. Find the probability density function of \(Y\).
  4. Sketch the probability density function of \(Y\).
  5. Write down the mode of \(Y\).
  6. Find \(\mathrm { E } ( Y )\).
  7. Find \(\mathrm { P } ( Y > 3 )\).
Edexcel S2 2009 June Q7
15 marks Moderate -0.8
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f3fdcd3c-c1c8-4205-a730-eb0bab8607d4-11_471_816_233_548} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the probability density function \(\mathrm { f } ( x )\) of the random variable \(X\). The part of the sketch from \(x = 0\) to \(x = 4\) consists of an isosceles triangle with maximum at ( \(2,0.5\) ).
  1. Write down \(\mathrm { E } ( X )\). The probability density function \(\mathrm { f } ( x )\) can be written in the following form. $$f ( x ) = \begin{cases} a x & 0 \leqslant x < 2 \\ b - a x & 2 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$
  2. Find the values of the constants \(a\) and \(b\).
  3. Show that \(\sigma\), the standard deviation of \(X\), is 0.816 to 3 decimal places.
  4. Find the lower quartile of \(X\).
  5. State, giving a reason, whether \(\mathrm { P } ( 2 - \sigma < X < 2 + \sigma )\) is more or less than 0.5
Edexcel S2 2009 June Q8
13 marks Standard +0.3
8. A cloth manufacturer knows that faults occur randomly in the production process at a rate of 2 every 15 metres.
  1. Find the probability of exactly 4 faults in a 15 metre length of cloth.
  2. Find the probability of more than 10 faults in 60 metres of cloth. A retailer buys a large amount of this cloth and sells it in pieces of length \(x\) metres. He chooses \(x\) so that the probability of no faults in a piece is 0.80
  3. Write down an equation for \(x\) and show that \(x = 1.7\) to 2 significant figures. The retailer sells 1200 of these pieces of cloth. He makes a profit of 60p on each piece of cloth that does not contain a fault but a loss of \(\pounds 1.50\) on any pieces that do contain faults.
  4. Find the retailer's expected profit.
Edexcel S2 2010 June Q3
5 marks Standard +0.3
3. A rectangle has a perimeter of 20 cm . The length, \(X \mathrm {~cm}\), of one side of this rectangle is uniformly distributed between 1 cm and 7 cm . Find the probability that the length of the longer side of the rectangle is more than 6 cm long.
Edexcel S2 2010 June Q5
15 marks Standard +0.3
  1. A company has a large number of regular users logging onto its website. On average 4 users every hour fail to connect to the company's website at their first attempt.
    1. Explain why the Poisson distribution may be a suitable model in this case.
    Find the probability that, in a randomly chosen 2 hour period,
    1. all users connect at their first attempt,
    2. at least 4 users fail to connect at their first attempt. The company suffered from a virus infecting its computer system. During this infection it was found that the number of users failing to connect at their first attempt, over a 12 hour period, was 60 .
  3. Using a suitable approximation, test whether or not the mean number of users per hour who failed to connect at their first attempt had increased. Use a \(5 \%\) level of significance and state your hypotheses clearly.