Questions — AQA (3508 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 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics Stats 1 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 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 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics 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
AQA S3 2013 June Q6
11 marks Standard +0.3
6 The demand for a WWSatNav at a superstore may be modelled by a Poisson distribution with a mean of 2.5 per day. The superstore is open 6 days each week, from Monday morning to Saturday evening.
    1. Determine the probability that the demand for WWSatNavs during a particular week is at most 18 .
    2. The superstore receives a delivery of WWSatNavs on each Sunday evening. The manager, Meena, requires that the probability of WWSatNavs being out of stock during a week should be at most \(5 \%\). Determine the minimum number of WWSatNavs that Meena requires to be in stock after a delivery.
    1. Use a distributional approximation to estimate the probability that the demand for WWSatNavs during a period of \(\mathbf { 2 }\) weeks is more than 35.
    2. Changes to the superstore's delivery schedule result in it receiving a delivery of WWSatNavs on alternate Sunday evenings. Meena now requires that the probability of WWSatNavs being out of stock during the 2 weeks following a delivery should be at most \(5 \%\). Use a distributional approximation to determine the minimum number of WWSatNavs that Meena now requires to be in stock after a delivery.
      (3 marks)
AQA S3 2013 June Q7
15 marks Standard +0.3
7 It is claimed that the proportion, \(P\), of people who prefer cooked fresh garden peas to cooked frozen garden peas is greater than 0.50 .
  1. In an attempt to investigate this claim, a sample of 50 people were each given an unlabelled portion of cooked fresh garden peas and an unlabelled portion of cooked frozen garden peas to taste. After tasting each portion, the people were each asked to state which of the two portions they preferred. Of the 50 people sampled, 29 preferred the cooked fresh garden peas. Assuming that the 50 people may be considered to constitute a random sample, use a binomial distribution and the \(10 \%\) level of significance to investigate the claim.
    (6 marks)
  2. It was then decided to repeat the tasting in part (a) but to involve a sample of 500 , rather than 50, people. Of the 500 people sampled, 271 preferred the cooked fresh garden peas.
    1. Assuming that the 500 people may be considered to constitute a random sample, use an approximation to the distribution of the sample proportion, \(\widehat { P }\), and the \(10 \%\) level of significance to again investigate the claim.
    2. The critical value of \(\widehat { P }\) for the test in part (b)(i) is 0.529 , correct to three significant figures. It is also given that, in fact, 55 per cent of people prefer cooked fresh garden peas. Estimate the power for a test of the claim that \(P > 0.50\) based on a random sample of 500 people and using the \(10 \%\) level of significance.
      (5 marks)
AQA S3 2014 June Q1
2 marks Moderate -0.8
1 A hotel's management is concerned about the quality of the free pens that it provides in its meeting rooms. The hotel's assistant manager tests a random sample of 200 such pens and finds that 23 of them fail to write immediately.
  1. Calculate an approximate \(\mathbf { 9 6 \% }\) confidence interval for the proportion of pens that fail to write immediately.
  2. The supplier of the pens to the hotel claims that at most 2 in 50 pens fail to write immediately. Comment, with numerical justification, on the supplier's claim.
    [0pt] [2 marks] QUESTION
    PART Answer space for question 1
AQA S3 2014 June Q2
6 marks Standard +0.3
2 Each household within a district council's area has two types of wheelie-bin: a black one for general refuse and a green one for garden refuse. Each type of bin is emptied by the council fortnightly. The weight, in kilograms, of refuse emptied from a black bin can be modelled by the random variable \(B \sim \mathrm {~N} \left( \mu _ { B } , 0.5625 \right)\). The weight, in kilograms, of refuse emptied from a green bin can be modelled by the random variable \(G \sim \mathrm {~N} \left( \mu _ { G } , 0.9025 \right)\). The mean weight of refuse emptied from a random sample of 20 black bins was 21.35 kg . The mean weight of refuse emptied from an independent random sample of 15 green bins was 21.90 kg . Test, at the \(5 \%\) level of significance, the hypothesis that \(\mu _ { B } = \mu _ { G }\).
[0pt] [6 marks]
AQA S3 2014 June Q3
12 marks Moderate -0.3
3 An investigation was carried out into the type of vehicle being driven when its driver was caught speeding. The investigation was restricted to drivers who were caught speeding when driving vehicles with at least 4 wheels. An analysis of the results showed that \(65 \%\) were driving cars ( C ), \(20 \%\) were driving vans (V) and 15\% were driving lorries (L). Of those driving cars, \(30 \%\) were caught by fixed speed cameras (F), 55\% were caught by mobile speed cameras (M) and 15\% were caught by average speed cameras (A). Of those driving vans, \(35 \%\) were caught by fixed speed cameras (F), \(45 \%\) were caught by mobile speed cameras (M) and 20\% were caught by average speed cameras (A). Of those driving lorries, \(10 \%\) were caught by fixed speed cameras \(( \mathrm { F } )\), \(65 \%\) were caught by mobile speed cameras (M) and \(25 \%\) were caught by average speed cameras (A).
  1. Represent this information by a tree diagram on which are shown labels and percentages or probabilities.
  2. Hence, or otherwise, calculate the probability that a driver, selected at random from those caught speeding:
    1. was driving either a car or a lorry and was caught by a mobile speed camera;
    2. was driving a lorry, given that the driver was caught by an average speed camera;
    3. was not caught by a fixed speed camera, given that the driver was not driving a car.
      [0pt] [8 marks]
  3. Three drivers were selected at random from those caught speeding by fixed speed cameras. Calculate the probability that they were driving three different types of vehicle.
    [0pt] [4 marks]
AQA S3 2014 June Q4
8 marks Moderate -0.3
4 A sample of 50 male Eastern Grey kangaroos had a mean weight of 42.6 kg and a standard deviation of 6.2 kg . A sample of 50 male Western Grey kangaroos had a mean weight of 39.7 kg and a standard deviation of 5.3 kg .
  1. Construct a 98\% confidence interval for the difference between the mean weight of male Eastern Grey kangaroos and that of male Western Grey kangaroos.
    [0pt] [5 marks]
    1. What assumption about the selection of each of the two samples was it necessary to make in order that the confidence interval constructed in part (a) was valid?
      [0pt] [1 mark]
    2. Why was it not necessary to assume anything about the distributions of the weights of male kangaroos in order that the confidence interval constructed in part (a) was valid?
      [0pt] [2 marks]
AQA S3 2014 June Q5
4 marks
5 The numbers of daily morning operations, \(X\), and daily afternoon operations, \(Y\), in an operating theatre of a small private hospital can be modelled by the following bivariate probability distribution.
\multirow{2}{*}{}Number of morning operations ( \(\boldsymbol { X }\) )
23456\(\mathbf { P } ( \boldsymbol { Y } = \boldsymbol { y } )\)
\multirow{3}{*}{Number of afternoon operations ( \(\boldsymbol { Y }\) )}30.000.050.200.200.050.50
40.000.150.100.050.000.30
50.050.050.100.000.000.20
\(\mathrm { P } ( \boldsymbol { X } = \boldsymbol { x } )\)0.050.250.400.250.051.00
    1. State why \(\mathrm { E } ( X ) = 4\) and show that \(\operatorname { Var } ( X ) = 0.9\).
    2. Given that $$\mathrm { E } ( Y ) = 3.7 , \operatorname { Var } ( Y ) = 0.61 \text { and } \mathrm { E } ( X Y ) = 14.4$$ calculate values for \(\operatorname { Cov } ( X , Y )\) and \(\rho _ { X Y }\).
  2. Calculate values for the mean and the variance of:
    1. \(T = X + Y\);
    2. \(\quad D = X - Y\).
      [0pt] [4 marks]
AQA S3 2014 June Q6
5 marks Standard +0.3
6 Population \(A\) has a normal distribution with unknown mean \(\mu _ { A }\) and a variance of 18.8.
Population \(B\) has a normal distribution with unknown mean \(\mu _ { B }\) but with the same variance as Population \(A\). The random variables \(\bar { X } _ { A }\) and \(\bar { X } _ { B }\) denote the means of independent samples, each of size \(n\), from populations \(A\) and \(B\) respectively.
  1. Find an expression, in terms of \(n\), for \(\operatorname { Var } \left( \bar { X } _ { A } - \bar { X } _ { B } \right)\).
  2. Given that the width of a \(99 \%\) confidence interval for \(\mu _ { A } - \mu _ { B }\) is to be at most 5 , calculate the minimum value for \(n\).
    [0pt] [5 marks]
AQA S3 2014 June Q7
4 marks Challenging +1.2
7
  1. The random variable \(X\) has a Poisson distribution with parameter \(\lambda\).
    1. Prove, from first principles, that \(\mathrm { E } ( X ) = \lambda\).
    2. Given that \(\mathrm { E } \left( X ^ { 2 } - X \right) = \lambda ^ { 2 }\), deduce that \(\operatorname { Var } ( X ) = \lambda\).
  2. The number of faults in a 100-metre ball of nylon string may be modelled by a Poisson distribution with parameter \(\lambda\).
    1. An analysis of one ball of string, selected at random, showed 15 faults. Using an exact test, investigate the claim that \(\lambda > 10\). Use the \(5 \%\) level of significance.
    2. A subsequent analysis of a random sample of 20 balls of string showed a total of 241 faults.
      (A) Using an approximate test, re-investigate the claim that \(\lambda > 10\). Use the \(5 \%\) level of significance.
      (B) Determine the critical value of the total number of faults for the test in part (b)(ii)(A).
      (C) Given that, in fact, \(\lambda = 12\), estimate the probability of a Type II error for a test of the claim that \(\lambda > 10\) based upon a random sample of 20 balls of string and using the \(5 \%\) level of significance.
      [0pt] [4 marks] \includegraphics[max width=\textwidth, alt={}, center]{d5852425-9340-4aae-82da-e3bf6772a0de-22_2490_1728_219_141}
      \includegraphics[max width=\textwidth, alt={}, center]{d5852425-9340-4aae-82da-e3bf6772a0de-23_2490_1719_217_150}
      \includegraphics[max width=\textwidth, alt={}, center]{d5852425-9340-4aae-82da-e3bf6772a0de-24_2489_1728_221_141}
AQA S3 2015 June Q1
6 marks Moderate -0.8
1 A demographer measured the length of the right foot, \(x\) millimetres, and the length of the right hand, \(y\) millimetres, of each of a sample of 12 males aged between 19 years and 25 years. The results are given in the table.
AQA S3 2015 June Q2
8 marks Standard +0.3
2 Emilia runs an online perfume business from home. She believes that she receives more orders on Mondays than on Fridays. She checked this during a period of 26 weeks and found that she received a total of 507 orders on the Mondays and a total of 416 orders on the Fridays. The daily numbers of orders that Emilia receives may be modelled by independent Poisson distributions with means \(\lambda _ { \mathrm { M } }\) for Mondays and \(\lambda _ { \mathrm { F } }\) for Fridays.
  1. Construct an approximate \(99 \%\) confidence interval for \(\lambda _ { \mathrm { M } } - \lambda _ { \mathrm { F } }\).
  2. Hence comment on Emilia's belief.
AQA S3 2015 June Q3
12 marks Moderate -0.3
3 A particular brand of spread is produced in three varieties: standard, light and very light. During a marketing campaign, the producer advertises that some cartons of spread contain coupons worth \(\pounds 1 , \pounds 2\) or \(\pounds 4\). For each variety of spread, the proportion of cartons containing coupons of each value is shown in the table.
\cline { 2 - 4 } \multicolumn{1}{c|}{}Variety
\cline { 2 - 4 } \multicolumn{1}{c|}{}StandardLightVery light
No coupon0.700.650.55
£1 coupon0.200.250.30
£2 coupon0.080.060.10
£4 coupon0.020.040.05
For example, the probability that a carton of standard spread contains a coupon worth \(\pounds 2\) is 0.08 . In a large batch of cartons, 55 per cent contain standard spread, 30 per cent contain light spread and 15 per cent contain very light spread.
  1. A carton of spread is selected at random from the batch. Find the probability that the carton:
    1. contains standard spread and a coupon worth \(\pounds 1\);
    2. does not contain a coupon;
    3. contains light spread, given that it does not contain a coupon;
    4. contains very light spread, given that it contains a coupon.
  2. A random sample of 3 cartons is selected from the batch. Given that all of these 3 cartons contain a coupon, find the probability that they each contain a different variety of spread.
    [0pt] [4 marks]
AQA S3 2015 June Q4
17 marks Moderate -0.3
4
  1. A large survey in the USA establishes that 60 per cent of its residents own a smartphone. A survey of 250 UK residents reveals that 164 of them own a smartphone.
    Assuming that these 250 UK residents may be regarded as a random sample, investigate the claim that the percentage of UK residents owning a smartphone is the same as that in the USA. Use the 5\% level of significance.
  2. A random sample of 40 residents in a market town reveals that 5 of them own a 4 G mobile phone. Use an exact test to investigate, at the \(5 \%\) level of significance, the belief that fewer than 25 per cent of the town's residents own a 4 G mobile phone.
  3. A marketing company needs to estimate the proportion of residents in a large city who own a 4 G mobile phone. It wishes to estimate this proportion to within 0.05 with a confidence of 98\%. Given that the proportion is known to be at most 30 per cent, estimate the sample size necessary in order to meet the company's need.
    [0pt] [5 marks]
AQA S3 2015 June Q5
16 marks Standard +0.3
5
  1. The random variable \(X\) has a binomial distribution with parameters \(n\) and \(p\).
    1. Prove, from first principles, that \(\mathrm { E } ( X ) = n p\).
    2. Given that \(\mathrm { E } ( X ( X - 1 ) ) = n ( n - 1 ) p ^ { 2 }\), find an expression for \(\operatorname { Var } ( X )\).
    1. The random variable \(Y\) has a binomial distribution with \(\mathrm { E } ( Y ) = 3\) and \(\operatorname { Var } ( Y ) = 2.985\). Find values for \(n\) and \(p\).
    2. The random variable \(U\) has \(\mathrm { E } ( U ) = 5\) and \(\operatorname { Var } ( U ) = 6.25\). Show that \(U\) does not have a binomial distribution.
  3. The random variable \(V\) has the distribution \(\operatorname { Po } ( 5 )\) and \(W = 2 V + 10\). Show that \(\mathrm { E } ( W ) = \operatorname { Var } ( W )\) but that \(W\) does not have a Poisson distribution.
  4. The probability that, in a particular country, a person has blood group AB negative is 0.2 per cent. A sample of 5000 people is selected. Given that the sample may be assumed to be random, use a distributional approximation to estimate the probability that at least 6 people but at most 12 people have blood group AB negative.
    [0pt] [3 marks]
AQA S3 2015 June Q6
16 marks Challenging +1.2
6
  1. The independent random variables \(S\) and \(L\) have means \(\mu _ { S }\) and \(\mu _ { L }\) respectively, and a common variance of \(\sigma ^ { 2 }\). The variable \(\bar { S }\) denotes the mean of a random sample of \(n\) observations on \(S\) and the variable \(\bar { L }\) denotes the mean of a random sample of \(n\) observations on \(L\). Find a simplified expression, in terms of \(\sigma ^ { 2 }\), for the variance of \(\bar { L } - 2 \bar { S }\).
  2. A machine fills both small bottles and large bottles with shower gel. It is known that the volume of shower gel delivered by the machine is normally distributed with a standard deviation of 8 ml .
    1. A random sample of 25 small bottles filled by the machine contained a mean volume of \(\bar { s } = 258 \mathrm { ml }\) of shower gel. An independent random sample of 25 large bottles filled by the machine contained a mean volume of \(\bar { l } = 522 \mathrm { ml }\) of shower gel. Investigate, at the \(10 \%\) level of significance, the hypothesis that the mean volume of shower gel in a large bottle is more than twice that in a small bottle.
      [0pt] [7 marks]
    2. Deduce that, for the test of the hypothesis in part (b)(i), the critical value of \(\bar { L } - 2 \bar { S }\) is 4.585 , correct to three decimal places.
      [0pt] [2 marks]
    3. In fact, the mean volume of shower gel in a large bottle exceeds twice that in a small bottle by 10 ml . Determine the probability of a Type II error for a test of the hypothesis in part (b)(i) at the 10\% level of significance, based upon random samples of 25 small bottles and 25 large bottles.
      [0pt] [4 marks]
AQA S3 2016 June Q1
8 marks Standard +0.3
1 In advance of a referendum on independence, the regional assembly of an eastern province of a particular country carried out an opinion poll to assess the strength of the 'Yes' vote. Of the 480 men polled, 264 indicated that they intended to vote 'Yes', and of the 500 women polled, 220 indicated that they intended to vote 'Yes'.
  1. Construct an approximate 95\% confidence interval for the difference between the proportion of men who intend to vote 'Yes' and the proportion of women who intend to vote 'Yes'.
  2. Comment on a claim that, in the forthcoming referendum, the percentage of men voting 'Yes' will exceed the percentage of women voting 'Yes' by at least 2.5 per cent. Justify your answer.
    [0pt] [2 marks]
    \includegraphics[max width=\textwidth, alt={}]{f536a1ad-333a-47ec-a076-ec8497c1d8fc-02_1380_1707_1327_153}
AQA S3 2016 June Q2
15 marks Moderate -0.3
2 A plane flies regularly between airports D and T with an intermediate stop at airport M . The time of the plane's departure from, or arrival at, each airport is classified as either early, on time, or late. On \(90 \%\) of flights, the plane departs from D on time, and on \(10 \%\) of flights, it departs from D late. Of those flights that depart from D on time, \(65 \%\) then depart from M on time and \(35 \%\) depart from M late. Of those flights that depart from D late, \(15 \%\) then depart from M on time and \(85 \%\) depart from M late. Any flight that departs from M on time has probability 0.25 of arriving at T early, probability 0.60 of arriving at T on time and probability 0.15 of arriving at T late. Any flight that departs from M late has probability 0.10 of arriving at T early, probability 0.20 of arriving at T on time and probability 0.70 of arriving at T late.
  1. Represent this information by a tree diagram on which labels and percentages or probabilities are shown.
  2. Hence, or otherwise, calculate the probability that the plane:
    1. arrives at T on time;
    2. arrives at T on time, given that it departed from D on time;
    3. does not arrive at T late, given that it departed from D on time;
    4. does not arrive at T late, given that it departed from M on time.
  3. Three independent flights of the plane depart from \(D\) on time. Calculate the probability that two flights arrive at T on time and that one flight arrives at T early.
    [0pt] [4 marks]
AQA S3 2016 June Q4
13 marks Standard +0.3
4 Ben is a fencing contractor who is often required to repair a garden fence by replacing a broken post between fence panels, as illustrated.
\includegraphics[max width=\textwidth, alt={}, center]{f536a1ad-333a-47ec-a076-ec8497c1d8fc-10_364_789_388_694} The tasks involved are as follows.
\(U :\) detach the two fence panels from the broken post
\(V\) : remove the broken post
\(W\) : insert a new post
\(X\) : attach the two fence panels to the new post
The mean and the standard deviation of the time, in minutes, for each of these tasks are shown in the table.
TaskMean
Standard
deviation
\(\boldsymbol { U }\)155
\(\boldsymbol { V }\)4015
\(\boldsymbol { W }\)7520
\(\boldsymbol { X }\)2010
The random variables \(U , V , W\) and \(X\) are pairwise independent, except for \(V\) and \(W\) for which \(\rho _ { V W } = 0.25\).
  1. Determine values for the mean and the variance of:
    1. \(R = U + X\);
    2. \(F = V + W\);
    3. \(T = R + F\);
    4. \(D = W - V\).
  2. Assuming that each of \(R , F , T\) and \(D\) is approximately normally distributed, determine the probability that:
    1. the total time taken by Ben to repair a garden fence is less than 3 hours;
    2. the time taken by Ben to insert a new post is at least 1 hour more than the time taken by him to remove the broken post.
AQA S3 2016 June Q5
10 marks Standard +0.3
5
  1. The random variable \(X\), which has distribution \(\mathrm { N } \left( \mu _ { X } , \sigma ^ { 2 } \right)\), is independent of the random variable \(Y\), which has distribution \(\mathrm { N } \left( \mu _ { Y } , \sigma ^ { 2 } \right)\). In order to test \(\mathrm { H } _ { 0 } : \mu _ { X } = 1.5 \mu _ { Y }\), samples of size \(n\) are taken on each of \(X\) and \(Y\) and the random variable \(\bar { D }\) is defined as $$\bar { D } = \bar { X } - 1.5 \bar { Y }$$ State the distribution of \(\bar { D }\) assuming that \(\mathrm { H } _ { 0 }\) is true.
  2. A machine that fills bags with rice delivers weights that are normally distributed with a standard deviation of 4.5 grams. The machine fills two sizes of bags: large and extra-large.
    The mean weight of rice in a random sample of 50 large bags is 1509 grams.
    The mean weight of rice in an independent random sample of 50 extra-large bags is 2261 grams. Test, at the \(5 \%\) level of significance, the claim that, on average, the rice in an extra-large bag is \(1 \frac { 1 } { 2 }\) times as heavy as that in a large bag.
    [0pt] [6 marks]
AQA S3 2016 June Q6
22 marks Challenging +1.2
6
  1. The discrete random variable \(X\) has probability distribution given by $$\mathrm { P } ( X = x ) = \begin{cases} \frac { \mathrm { e } ^ { - \lambda } \lambda ^ { x } } { x ! } & x = 0,1,2 , \ldots \\ 0 & \text { otherwise } \end{cases}$$ Show that \(\mathrm { E } ( X ) = \lambda\) and that \(\operatorname { Var } ( X ) = \lambda\).
  2. In light-weight chain, faults occur randomly and independently, and at a constant average rate of 0.075 per metre.
    1. Calculate the probability that there are no faults in a 10 -metre length of this chain.
    2. Use a distributional approximation to estimate the probability that, in a 500 -metre reel of light-weight chain, there are:
      (A) fewer than 30 faults;
      (B) at least 35 faults but at most 45 faults.
  3. As part of an investigation into the quality of a new design of medium-weight chain, a sample of fifty 10 -metre lengths was selected. Subsequent analysis revealed a total of 49 faults.
    Assuming that faults occur randomly and independently, and at a constant average rate, construct an approximate \(98 \%\) confidence interval for the average number of faults per metre.
    [0pt] [6 marks] \section*{DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED}
AQA M1 2005 January Q1
7 marks Moderate -0.8
1 A train travels along a straight horizontal track. It is travelling at a speed of \(12 \mathrm {~ms} ^ { - 1 }\) when it begins to accelerate uniformly. It reaches a speed of \(40 \mathrm {~ms} ^ { - 1 }\) after accelerating for 100 seconds.
    1. Show that the acceleration of the train is \(0.28 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
    2. Find the distance that the train travelled in the 100 seconds.
  2. The mass of the train is 200 tonnes and a resistance force of 40000 N acts on the train. Find the magnitude of the driving force produced by the engine that acts on the train as it accelerates.
AQA M1 2005 January Q2
7 marks Moderate -0.8
2 A particle, \(A\), of mass 12 kg is moving on a smooth horizontal surface with velocity \(\left[ \begin{array} { l } 4 \\ 7 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\). It then collides and coalesces with a second particle, \(B\), of mass 4 kg .
  1. If before the collision the velocity of \(B\) was \(\left[ \begin{array} { l } 2 \\ 3 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\), find the velocity of the combined particle after the collision.
  2. If after the collision the velocity of the combined particle is \(\left[ \begin{array} { l } 1 \\ 4 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\), find the velocity of \(B\) before the collision.
AQA M1 2005 January Q3
11 marks Moderate -0.3
3 The diagram shows a rope that is attached to a box of mass 25 kg , which is being pulled along rough horizontal ground. The rope is at an angle of \(30 ^ { \circ }\) to the ground. The tension in the rope is 40 N . The box accelerates at \(0.1 \mathrm {~ms} ^ { - 2 }\).
\includegraphics[max width=\textwidth, alt={}, center]{eb1f2470-aeeb-4b1d-a6c0-bdeb7048edd5-3_214_729_504_644}
  1. Draw a diagram to show all of the forces acting on the box.
  2. Show that the magnitude of the friction force acting on the box is 32.1 N , correct to three significant figures.
  3. Show that the magnitude of the normal reaction force that the ground exerts on the box is 225 N .
  4. Find the coefficient of friction between the box and the ground.
  5. State what would happen to the magnitude of the friction force if the angle between the rope and the horizontal were increased. Give a reason for your answer.
AQA M1 2005 January Q4
8 marks Moderate -0.8
4 Two particles are connected by a string, which passes over a pulley. Model the string as light and inextensible. The particles have masses of 2 kg and 5 kg . The particles are released from rest.
\includegraphics[max width=\textwidth, alt={}, center]{eb1f2470-aeeb-4b1d-a6c0-bdeb7048edd5-3_392_209_1685_909}
  1. State one modelling assumption that you should make about the pulley in order to determine the acceleration of the particles.
  2. By forming an equation of motion for each particle, show that the magnitude of the acceleration of each particle is \(4.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  3. Find the tension in the string.
AQA M1 2005 January Q5
7 marks Moderate -0.8
5 Two ropes are attached to a load of mass 500 kg . The ropes make angles of \(30 ^ { \circ }\) and \(45 ^ { \circ }\) to the vertical, as shown in the diagram. The tensions in these ropes are \(T _ { 1 }\) and \(T _ { 2 }\) newtons. The load is also supported by a vertical spring.
\includegraphics[max width=\textwidth, alt={}, center]{eb1f2470-aeeb-4b1d-a6c0-bdeb7048edd5-4_533_565_507_744} The system is in equilibrium and \(T _ { 1 } = 200\).
  1. Show that \(T _ { 2 } = 141\), correct to three significant figures.
  2. Find the force that the spring exerts on the load.