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AQA C4 2009 June Q3
13 marks Moderate -0.3
3
  1. Find the binomial expansion of \(( 1 - x ) ^ { - 1 }\) up to and including the term in \(x ^ { 2 }\).
    1. Express \(\frac { 3 x - 1 } { ( 1 - x ) ( 2 - 3 x ) }\) in the form \(\frac { A } { 1 - x } + \frac { B } { 2 - 3 x }\), where \(A\) and \(B\) are integers.
    2. Find the binomial expansion of \(\frac { 3 x - 1 } { ( 1 - x ) ( 2 - 3 x ) }\) up to and including the term in \(x ^ { 2 }\).
  3. Find the range of values of \(x\) for which the binomial expansion of \(\frac { 3 x - 1 } { ( 1 - x ) ( 2 - 3 x ) }\) is valid.
AQA C4 2009 June Q4
6 marks Moderate -0.3
4 A car depreciates in value according to the model $$V = A k ^ { t }$$ where \(\pounds V\) is the value of the car \(t\) months from when it was new, and \(A\) and \(k\) are constants. Its value when new was \(\pounds 12499\) and 36 months later its value was \(\pounds 7000\).
    1. Write down the value of \(A\).
    2. Show that the value of \(k\) is 0.984025 , correct to six decimal places.
  2. The value of this car first dropped below \(\pounds 5000\) during the \(n\)th month from new. Find the value of \(n\).
AQA C4 2009 June Q5
5 marks Standard +0.3
5 A curve is defined by the equation \(4 x ^ { 2 } + y ^ { 2 } = 4 + 3 x y\).
Find the gradient at the point ( 1,3 ) on this curve.
AQA C4 2009 June Q6
15 marks Standard +0.3
6
    1. Show that the equation \(3 \cos 2 x + 7 \cos x + 5 = 0\) can be written in the form \(a \cos ^ { 2 } x + b \cos x + c = 0\), where \(a , b\) and \(c\) are integers.
    2. Hence find the possible values of \(\cos x\).
    1. Express \(7 \sin \theta + 3 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(\alpha\) is an acute angle. Give your value of \(\alpha\) to the nearest \(0.1 ^ { \circ }\).
    2. Hence solve the equation \(7 \sin \theta + 3 \cos \theta = 4\) for all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\), giving \(\theta\) to the nearest \(0.1 ^ { \circ }\).
    1. Given that \(\beta\) is an acute angle and that \(\tan \beta = 2 \sqrt { 2 }\), show that \(\cos \beta = \frac { 1 } { 3 }\).
    2. Hence show that \(\sin 2 \beta = p \sqrt { 2 }\), where \(p\) is a rational number.
AQA C4 2009 June Q7
10 marks Moderate -0.3
7 The points \(A\) and \(B\) have coordinates ( \(3 , - 2,5\) ) and ( \(4,0,1\) ) respectively. The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left[ \begin{array} { r } 6 \\ - 1 \\ 5 \end{array} \right] + \lambda \left[ \begin{array} { r } 2 \\ - 1 \\ 4 \end{array} \right]\).
  1. Find the distance between the points \(A\) and \(B\).
  2. Verify that \(B\) lies on \(l _ { 1 }\).
    (2 marks)
  3. The line \(l _ { 2 }\) passes through \(A\) and has equation \(\mathbf { r } = \left[ \begin{array} { r } 3 \\ - 2 \\ 5 \end{array} \right] + \mu \left[ \begin{array} { r } - 1 \\ 3 \\ - 8 \end{array} \right]\). The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(C\). Show that the points \(A , B\) and \(C\) form an isosceles triangle.
    (6 marks)
AQA C4 2009 June Q8
10 marks Moderate -0.3
8
  1. Solve the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 150 \cos 2 t } { x }$$ given that \(x = 20\) when \(t = \frac { \pi } { 4 }\), giving your solution in the form \(x ^ { 2 } = \mathrm { f } ( t )\). (6 marks)
  2. The oscillations of a 'baby bouncy cradle' are modelled by the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 150 \cos 2 t } { x }$$ where \(x \mathrm {~cm}\) is the height of the cradle above its base \(t\) seconds after the cradle begins to oscillate. Given that the cradle is 20 cm above its base at time \(t = \frac { \pi } { 4 }\) seconds, find:
    1. the height of the cradle above its base 13 seconds after it starts oscillating, giving your answer to the nearest millimetre;
    2. the time at which the cradle will first be 11 cm above its base, giving your answer to the nearest tenth of a second.
      (2 marks)
AQA S1 2005 January Q1
7 marks Moderate -0.3
1 Each Monday, Azher has a stall at a town's outdoor market. The table below shows, for each of a random sample of 10 Mondays during 2003, the air temperature, \(x ^ { \circ } \mathrm { C }\), at 9 am and Azher's takings, £y.
Monday\(\mathbf { 1 }\)\(\mathbf { 2 }\)\(\mathbf { 3 }\)\(\mathbf { 4 }\)\(\mathbf { 5 }\)\(\mathbf { 6 }\)\(\mathbf { 7 }\)\(\mathbf { 8 }\)\(\mathbf { 9 }\)\(\mathbf { 1 0 }\)
\(\boldsymbol { x }\)2691813712134
\(\boldsymbol { y }\)9710313624512178145128141312
  1. A scatter diagram of these data is shown below. \includegraphics[max width=\textwidth, alt={}, center]{7faa4a2d-f5cc-4cc3-a3a9-5d8290ceabdc-2_901_1068_1078_447} Give two distinct comments, in context, on what this diagram reveals.
  2. One of the Mondays is found to be Easter Monday, the busiest Monday market of the year. Identify which Monday this is most likely to be.
  3. Removing the data for the Monday you identified in part (b), calculate the value of the product moment correlation coefficient for the remaining 9 pairs of values of \(x\) and \(y\).
  4. Name one other variable that would have been likely to affect Azher's takings at this town's outdoor market.
    (l mark)
AQA S1 2005 January Q2
9 marks Moderate -0.3
2 The volume, in millilitres, of lemonade in mini-cans may be assumed to be normally distributed with a standard deviation of 3.5. The volumes, in millilitres, of lemonade in a random sample of 12 mini-cans were as follows.
155148156149147156
157156150154148154
  1. Construct a \(98 \%\) confidence interval for the mean volume of lemonade in a mini-can, giving the limits to one decimal place.
  2. On each mini-can is printed " 150 ml ". Comment on this, using the given sample and your confidence interval in part (a).
  3. State why, in part (a), use of the Central Limit Theorem was not necessary.
AQA S1 2005 January Q3
12 marks Moderate -0.8
3 [Figure 1, printed on the insert, is provided for use in this question.]
A parcel delivery company has a depot on the outskirts of a town. Each weekday, a van leaves the depot to deliver parcels across a nearby area. The table below shows, for a random sample of 10 weekdays, the number, \(x\), of parcels to be delivered and the total time, \(y\) minutes, that the van is out of the depot.
\(\boldsymbol { x }\)9162211192614101117
\(\boldsymbol { y }\)791271721091522141318094148
  1. On Figure 1, plot a scatter diagram of these data.
  2. Calculate the equation of the least squares regression line of \(y\) on \(x\) and draw your line on Figure 1.
  3. Use your regression equation to estimate the total time that the van is out of the depot when delivering:
    1. 15 parcels;
    2. 35 parcels. Comment on the likely reliability of each of your estimates.
  4. The time that the van is out of the depot delivering parcels may be thought of as the time needed to travel to and from the area plus an amount of time proportional to the number of parcels to be delivered. Given that the regression line of \(y\) on \(x\) is of the form \(y = a + b x\), give an interpretation, in context, for each of your values of \(a\) and \(b\).
    (2 marks)
AQA S1 2005 January Q4
15 marks Moderate -0.3
4 Chopped lettuce is sold in bags nominally containing 100 grams.
The weight, \(X\) grams, of chopped lettuce, delivered by the machine filling the bags, may be assumed to be normally distributed with mean \(\mu\) and standard deviation 4.
  1. Assuming that \(\mu = 106\), determine the probability that a randomly selected bag of chopped lettuce:
    1. weighs less than 110 grams;
    2. is underweight.
  2. Determine the minimum value of \(\mu\) so that at most 2 per cent of bags of chopped lettuce are underweight. Give your answer to one decimal place.
  3. Boxes each contain 10 bags of chopped lettuce. The mean weight of a bag of chopped lettuce in a box is denoted by \(\bar { X }\). Given that \(\mu = 108.5\) :
    1. write down values for the mean and variance of \(\bar { X }\);
    2. determine the probability that \(\bar { X }\) exceeds 110 .
AQA S1 2005 January Q5
15 marks Standard +0.3
5 Each evening Aaron sets his alarm for 7 am. He believes that the probability that he wakes before his alarm rings each morning is 0.4 , and is independent from morning to morning.
  1. Assuming that Aaron's belief is correct, determine the probability that, during a week (7 mornings), he wakes before his alarm rings:
    1. on 2 or fewer mornings;
    2. on more than 1 but fewer than 5 mornings.
  2. Assuming that Aaron's belief is correct, calculate the probability that, during a 4 -week period, he wakes before his alarm rings on exactly 7 mornings.
  3. Assuming that Aaron's belief is correct, calculate values for the mean and standard deviation of the number of mornings in a week when Aaron wakes before his alarm rings.
    (2 marks)
  4. During a 50-week period, Aaron records, each week, the number of mornings on which he wakes before his alarm rings. The results are as follows.
    Number of mornings01234567
    Frequency108775544
    1. Calculate the mean and standard deviation of these data.
    2. State, giving reasons, whether your answers to part (d)(i) support Aaron's belief that the probability that he wakes before his alarm rings each morning is 0.4 , and is independent from morning to morning.
      (3 marks)
AQA S1 2005 January Q6
14 marks Easy -1.2
6 The table below shows the numbers of males and females in each of three employment categories at a university on 31 July 2003.
\cline { 2 - 4 } \multicolumn{1}{c|}{}Employment category
\cline { 2 - 4 } \multicolumn{1}{c|}{}ManagerialAcademicSupport
Male38369303
Female26275643
  1. An employee is selected at random. Determine the probability that the employee is:
    1. female;
    2. a female academic;
    3. either female or academic or both;
    4. female, given that the employee is academic.
  2. Three employees are selected at random, without replacement. Determine the probability that:
    1. all three employees are male;
    2. exactly one employee is male.
  3. The event "employee selected is academic" is denoted by \(A\). The event "employee selected is female" is denoted by \(F\). Describe in context, as simply as possible, the events denoted by:
    1. \(F \cap A\);
    2. \(F ^ { \prime } \cup A\).
      SurnameOther Names
      Centre NumberCandidate Number
      Candidate Signature
      General Certificate of Education
      January 2005
      Advanced Subsidiary Examination MS/SS1B AQA
      459:5EMLM
      : 11 P וPII " 1 : : ר
      ALLI.ub c \section*{STATISTICS} Unit Statistics 1B Insert for use in Question 3.
      Fill in the boxes at the top of this page.
      Fasten this insert securely to your answer book. \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Scatter diagram for parcel deliveries by a van} \includegraphics[alt={},max width=\textwidth]{7faa4a2d-f5cc-4cc3-a3a9-5d8290ceabdc-8_2420_1664_349_175}
      \end{figure} Figure 1 (for Question 3)
AQA S1 2007 January Q1
9 marks Easy -1.2
1 The times, in seconds, taken by 20 people to solve a simple numerical puzzle were
17192226283134363839
41424347505153555758
  1. Calculate the mean and the standard deviation of these times.
  2. In fact, 23 people solved the puzzle. However, 3 of them failed to solve it within the allotted time of 60 seconds. Calculate the median and the interquartile range of the times taken by all 23 people.
    (4 marks)
  3. For the times taken by all 23 people, explain why:
    1. the mode is not an appropriate numerical measure;
    2. the range is not an appropriate numerical measure.
AQA S1 2007 January Q2
12 marks Moderate -0.8
2 A hotel has 50 single rooms, 16 of which are on the ground floor. The hotel offers guests a choice of a full English breakfast, a continental breakfast or no breakfast. The probabilities of these choices being made are \(0.45,0.25\) and 0.30 respectively. It may be assumed that the choice of breakfast is independent from guest to guest.
  1. On a particular morning there are 16 guests, each occupying a single room on the ground floor. Calculate the probability that exactly 5 of these guests require a full English breakfast.
  2. On a particular morning when there are 50 guests, each occupying a single room, determine the probability that:
    1. at most 12 of these guests require a continental breakfast;
    2. more than 10 but fewer than 20 of these guests require no breakfast.
  3. When there are 40 guests, each occupying a single room, calculate the mean and the standard deviation for the number of these guests requiring breakfast.
AQA S1 2007 January Q3
5 marks Easy -1.3
3 Estimate, without undertaking any calculations, the value of the product moment correlation coefficient between the variables \(x\) and \(y\) in each of the three scatter diagrams.
  1. \includegraphics[max width=\textwidth, alt={}, center]{868dc38b-3f24-4218-a300-c3cc2d9ff5d1-03_631_659_516_301}
  2. \includegraphics[max width=\textwidth, alt={}, center]{868dc38b-3f24-4218-a300-c3cc2d9ff5d1-03_620_647_525_1119}
  3. \includegraphics[max width=\textwidth, alt={}, center]{868dc38b-3f24-4218-a300-c3cc2d9ff5d1-03_624_655_1279_303}
    (5 marks)
AQA S1 2007 January Q4
7 marks Moderate -0.3
4 A very popular play has been performed at a London theatre on each of 6 evenings per week for about a year. Over the past 13 weeks ( 78 performances), records have been kept of the proceeds from the sales of programmes at each performance. An analysis of these records has found that the mean was \(\pounds 184\) and the standard deviation was \(\pounds 32\).
  1. Assuming that the 78 performances may be considered to be a random sample, construct a \(90 \%\) confidence interval for the mean proceeds from the sales of programmes at an evening performance of this play.
  2. Comment on the likely validity of the assumption in part (a) when constructing a confidence interval for the mean proceeds from the sales of programmes at an evening performance of:
    1. this particular play;
    2. any play.
AQA S1 2007 January Q5
10 marks Moderate -0.8
5 Dafydd, Eli and Fabio are members of an amateur cycling club that holds a time trial each Sunday during the summer. The independent probabilities that Dafydd, Eli and Fabio take part in any one of these trials are \(0.6,0.7\) and 0.8 respectively. Find the probability that, on a particular Sunday during the summer:
  1. none of the three cyclists takes part;
  2. Fabio is the only one of the three cyclists to take part;
  3. exactly one of the three cyclists takes part;
  4. either one or two of the three cyclists take part.
AQA S1 2007 January Q6
17 marks Moderate -0.3
6 When Monica walks to work from home, she uses either route A or route B.
  1. Her journey time, \(X\) minutes, by route A may be assumed to be normally distributed with a mean of 37 and a standard deviation of 8 . Determine:
    1. \(\mathrm { P } ( X < 45 )\);
    2. \(\mathrm { P } ( 30 < X < 45 )\).
  2. Her journey time, \(Y\) minutes, by route B may be assumed to be normally distributed with a mean of 40 and a standard deviation of \(\sigma\). Given that \(\mathrm { P } ( Y > 45 ) = 0.12\), calculate the value of \(\sigma\).
  3. If Monica leaves home at 8.15 am to walk to work hoping to arrive by 9.00 am , state, with a reason, which route she should take.
  4. When Monica travels to work from home by car, her journey time, \(W\) minutes, has a mean of 18 and a standard deviation of 12 . Estimate the probability that, for a random sample of 36 journeys to work from home by car, Monica's mean time is more than 20 minutes.
  5. Indicate where, if anywhere, in this question you needed to make use of the Central Limit Theorem.
AQA S1 2007 January Q7
15 marks Moderate -0.8
7 [Figure 1, printed on the insert, is provided for use in this question.]
Stan is a retired academic who supplements his pension by mowing lawns for customers who live nearby. As part of a review of his charges for this work, he measures the areas, \(x \mathrm {~m} ^ { 2 }\), of a random sample of eight of his customers' lawns and notes the times, \(y\) minutes, that it takes him to mow these lawns. His results are shown in the table.
Customer\(\mathbf { A }\)\(\mathbf { B }\)\(\mathbf { C }\)\(\mathbf { D }\)\(\mathbf { E }\)\(\mathbf { F }\)\(\mathbf { G }\)\(\mathbf { H }\)
\(\boldsymbol { x }\)3601408606001180540260480
\(\boldsymbol { y }\)502513570140905570
  1. On Figure 1, plot a scatter diagram of these data.
  2. Calculate the equation of the least squares regression line of \(y\) on \(x\). Draw your line on Figure 1.
  3. Calculate the value of the residual for Customer H and indicate how your value is confirmed by your scatter diagram.
  4. Given that Stan charges \(\pounds 12\) per hour, estimate the charge for mowing a customer's lawn that has an area of \(560 \mathrm {~m} ^ { 2 }\).
AQA S1 2010 January Q1
9 marks Moderate -0.8
1 Draught excluder for doors and windows is sold in rolls of nominal length 10 metres.
The actual length, \(X\) metres, of draught excluder on a roll may be modelled by a normal distribution with mean 10.2 and standard deviation 0.15 .
  1. Determine:
    1. \(\mathrm { P } ( X < 10.5 )\);
    2. \(\mathrm { P } ( 10.0 < X < 10.5 )\).
  2. A customer randomly selects six 10 -metre rolls of the draught excluder. Calculate the probability that all six rolls selected contain more than 10 metres of draught excluder.
AQA S1 2010 January Q2
8 marks Moderate -0.8
2 Lizzie, the receptionist at a dental practice, was asked to keep a weekly record of the number of patients who failed to turn up for an appointment. Her records for the first 15 weeks were as follows. $$\begin{array} { l l l l l l l l l l l l l l l } 20 & 26 & 32 & a & 37 & 14 & 27 & 34 & 15 & 18 & b & 25 & 37 & 29 & 25 \end{array}$$ Unfortunately, Lizzie forgot to record the actual values for two of the 15 weeks, so she recorded them as \(a\) and \(b\). However, she did remember that \(a < 10\) and that \(b > 40\).
  1. Calculate the median and the interquartile range of these 15 values.
  2. Give a reason why, for these data:
    1. the mode is not an appropriate measure of average;
    2. the standard deviation cannot be used as a measure of spread.
  3. Subsequent investigations revealed that the missing values were 8 and 43 . Calculate the mean and the standard deviation of the 15 values.
AQA S1 2010 January Q3
8 marks Moderate -0.3
3 The table shows, for each of a random sample of 7 weeks, the number of customers, \(x\), who purchased fuel from a filling station, together with the total volume, \(y\) litres, of fuel purchased by these customers.
\(\boldsymbol { x }\)230184165147241174210
\(\boldsymbol { y }\)4551341032523756378740244254
  1. Calculate the equation of the least squares regression line of \(y\) on \(x\).
  2. Estimate the volume of fuel sold during a week in which 200 customers purchase fuel.
  3. Comment on the likely reliability of your estimate in part (b), given that, for the regression line calculated in part (a), the values of the 7 residuals lie between approximately - 415 litres and + 430 litres.
AQA S1 2010 January Q4
12 marks Moderate -0.8
4 Each school-day morning, three students, Rita, Said and Ting, travel independently from their homes to the same school by one of three methods: walk, cycle or bus. The table shows the probabilities of their independent daily choices.
\cline { 2 - 4 } \multicolumn{1}{c|}{}WalkCycleBus
Rita0.650.100.25
Said0.400.450.15
Ting0.250.550.20
  1. Calculate the probability that, on any given school-day morning:
    1. all 3 students walk to school;
    2. only Rita travels by bus to school;
    3. at least 2 of the 3 students cycle to school.
  2. Ursula, a friend of Rita, never travels to school by bus. The probability that: Ursula walks to school when Rita walks to school is 0.9 ; Ursula cycles to school when Rita cycles to school is 0.7 . Calculate the probability that, on any given school-day morning, Rita and Ursula travel to school by:
    1. the same method;
    2. different methods.
AQA S1 2010 January Q5
11 marks Moderate -0.3
5 In a random sample of 12 bags of flour, the weight, in grams, of flour in each bag was recorded as follows. \(\begin{array} { l l l l l l l l l l l l } 1011 & 995 & 1018 & 1022 & 1014 & 1005 & 1017 & 1015 & 993 & 1018 & 992 & 1020 \end{array}\)
  1. It may be assumed that the weight of flour in a bag is normally distributed with a standard deviation of 10.5 grams.
    1. Construct a \(98 \%\) confidence interval for the mean weight, \(\mu\) grams, of flour in a bag, giving the limits to four significant figures.
    2. State why, in constructing your confidence interval, use of the Central Limit Theorem was not necessary.
    3. If the distribution of the weight of flour in a bag was unknown, indicate a minimum number of weights that you would consider necessary for a confidence interval for \(\mu\) to be valid.
  2. The statement ' 1 kg ' is printed on each bag. Comment on this statement using both the confidence interval that you constructed in part (a)(i) and the weights of the given sample of 12 bags.
  3. Given that \(\mu = 1000\), state the probability that a \(98 \%\) confidence interval for \(\mu\) will not contain 1000.
    (l mark)
AQA S1 2010 January Q6
14 marks Moderate -0.3
6 During the winter, the probability that Barry's cat, Sylvester, chooses to stay outside all night is 0.35 , and the cat's choice is independent from night to night.
  1. Determine the probability that, during a period of 2 weeks ( 14 nights) in winter, Sylvester chooses to stay outside:
    1. on at most 7 nights;
    2. on at least 11 nights;
    3. on more than 5 nights but fewer than 10 nights.
  2. Calculate the probability that, during a period of \(\mathbf { 3 }\) weeks in winter, Sylvester chooses to stay outside on exactly 4 nights.
  3. Barry claims that, during the summer, the number of nights per week, \(S\), on which Sylvester chooses to stay outside can be modelled by a binomial distribution with \(n = 7\) and \(p = \frac { 5 } { 7 }\).
    1. Assuming that Barry's claim is correct, find the mean and the variance of \(S\).
    2. For a period of 13 weeks during the summer, the number of nights per week on which Sylvester chose to stay outside had a mean of 5 and a variance of 1.5 . Comment on Barry's claim.
      (2 marks)