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AQA S2 Q1
1 A study undertaken by Goodhealth Hospital found that the number of patients each month, \(X\), contracting a particular superbug can be modelled by a Poisson distribution with a mean of 1.5 .
    1. Calculate \(\mathrm { P } ( X = 2 )\).
    2. Hence determine the probability that exactly 2 patients will contract this superbug in each of three consecutive months.
    1. Write down the distribution of \(Y\), the number of patients contracting this superbug in a given 6-month period.
    2. Find the probability that at least 12 patients will contract this superbug during a given 6-month period.
  3. State two assumptions implied by the use of a Poisson model for the number of patients contracting this superbug.
AQA S2 Q2
2 Year 12 students at Newstatus School choose to participate in one of four sports during the Spring term. The students' choices are summarised in the table.
SquashBadmintonArcheryHockeyTotal
Male516301970
Female4203353110
Total9366372180
  1. Use a \(\chi ^ { 2 }\) test, at the \(5 \%\) level of significance, to determine whether the choice of sport is independent of gender.
  2. Interpret your result in part (a) as it relates to students choosing hockey.
AQA S2 Q3
3 The time, \(T\) minutes, that parents have to wait before seeing a mathematics teacher at a school parents' evening can be modelled by a normal distribution with mean \(\mu\) and standard deviation \(\sigma\). At a recent parents' evening, a random sample of 9 parents was asked to record the times that they waited before seeing a mathematics teacher. The times, in minutes, are $$\begin{array} { l l l l l l l l l } 5 & 12 & 10 & 8 & 7 & 6 & 9 & 7 & 8 \end{array}$$
  1. Construct a \(90 \%\) confidence interval for \(\mu\).
  2. Comment on the headteacher's claim that the mean time that parents have to wait before seeing a mathematics teacher is 5 minutes.
AQA S2 Q4
4
  1. A random variable \(X\) has probability density function defined by $$\mathrm { f } ( x ) = \begin{cases} k & a < x < b \\ 0 & \text { otherwise } \end{cases}$$
    1. Show that \(k = \frac { 1 } { b - a }\).
    2. Prove, using integration, that \(\mathrm { E } ( X ) = \frac { 1 } { 2 } ( a + b )\).
  2. The error, \(X\) grams, made when a shopkeeper weighs out loose sweets can be modelled by a rectangular distribution with the following probability density function: $$f ( x ) = \begin{cases} k & - 2 < x < 4 \\ 0 & \text { otherwise } \end{cases}$$
    1. Write down the value of the mean, \(\mu\), of \(X\).
    2. Evaluate the standard deviation, \(\sigma\), of \(X\).
    3. Hence find \(\mathrm { P } \left( X < \frac { 2 - \mu } { \sigma } \right)\).
AQA S2 Q5
5 The Globe Express agency organises trips to the theatre. The cost, \(\pounds X\), of these trips can be modelled by the following probability distribution:
\(\boldsymbol { x }\)40455574
\(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)0.300.240.360.10
  1. Calculate the mean and standard deviation of \(X\).
  2. For special celebrity charity performances, Globe Express increases the cost of the trips to \(\pounds Y\), where $$Y = 10 X + 250$$ Determine the mean and standard deviation of \(Y\).
AQA S2 Q6
6 In previous years, the marks obtained in a French test by students attending Topnotch College have been modelled satisfactorily by a normal distribution with a mean of 65 and a standard deviation of 9 . Teachers in the French department at Topnotch College suspect that this year their students are, on average, underachieving. In order to investigate this suspicion, the teachers selected a random sample of 35 students to take the French test and found that their mean score was 61.5.
  1. Investigate, at the \(5 \%\) level of significance, the teachers' suspicion.
  2. Explain, in the context of this question, the meaning of a Type I error.
AQA S2 Q7
7 Engineering work on the railway network causes an increase in the journey time of commuters travelling into work each morning. The increase in journey time, \(T\) hours, is modelled by a continuous random variable with probability density function $$\mathrm { f } ( t ) = \begin{cases} 4 t \left( 1 - t ^ { 2 } \right) & 0 \leqslant t \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$
  1. Show that \(\mathrm { E } ( T ) = \frac { 8 } { 15 }\).
    1. Find the cumulative distribution function, \(\mathrm { F } ( t )\), for \(0 \leqslant t \leqslant 1\).
    2. Hence, or otherwise, for a commuter selected at random, find $$\mathrm { P } ( \text { mean } < T < \text { median } )$$
AQA S2 Q8
8 Bottles of sherry nominally contain 1000 millilitres. After the introduction of a new method of filling the bottles, there is a suspicion that the mean volume of sherry in a bottle has changed. In order to investigate this suspicion, a random sample of 12 bottles of sherry is taken and the volume of sherry in each bottle is measured. The volumes, in millilitres, of sherry in these bottles are found to be
9961006100999910071003
998101099799610081007
Assuming that the volume of sherry in a bottle is normally distributed, investigate, at the \(5 \%\) level of significance, whether the mean volume of sherry in a bottle differs from 1000 millilitres.
AQA FP2 Q1
1
  1. Show that $$\frac { 1 } { r ^ { 2 } } - \frac { 1 } { ( r + 1 ) ^ { 2 } } = \frac { 2 r + 1 } { r ^ { 2 } ( r + 1 ) ^ { 2 } }$$
  2. Hence find the sum of the first \(n\) terms of the series $$\frac { 3 } { 1 ^ { 2 } \times 2 ^ { 2 } } + \frac { 5 } { 2 ^ { 2 } \times 3 ^ { 2 } } + \frac { 7 } { 3 ^ { 2 } \times 4 ^ { 2 } } + \ldots$$
AQA FP2 Q2
2 The cubic equation $$x ^ { 3 } + p x ^ { 2 } + q x + r = 0$$ where \(p , q\) and \(r\) are real, has roots \(\alpha , \beta\) and \(\gamma\).
  1. Given that $$\alpha + \beta + \gamma = 4 \quad \text { and } \quad \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 20$$ find the values of \(p\) and \(q\).
  2. Given further that one root is \(3 + \mathrm { i }\), find the value of \(r\).
AQA FP2 Q3
3 The complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) are given by $$z _ { 1 } = \frac { 1 + \mathrm { i } } { 1 - \mathrm { i } } \quad \text { and } \quad z _ { 2 } = \frac { 1 } { 2 } + \frac { \sqrt { 3 } } { 2 } \mathrm { i }$$
  1. Show that \(z _ { 1 } = \mathrm { i }\).
  2. Show that \(\left| z _ { 1 } \right| = \left| z _ { 2 } \right|\).
  3. Express both \(z _ { 1 }\) and \(z _ { 2 }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
  4. Draw an Argand diagram to show the points representing \(z _ { 1 } , z _ { 2 }\) and \(z _ { 1 } + z _ { 2 }\).
  5. Use your Argand diagram to show that $$\tan \frac { 5 } { 12 } \pi = 2 + \sqrt { 3 }$$
AQA FP2 Q4
4
  1. Prove by induction that $$2 + ( 3 \times 2 ) + \left( 4 \times 2 ^ { 2 } \right) + \ldots + ( n + 1 ) 2 ^ { n - 1 } = n 2 ^ { n }$$ for all integers \(n \geqslant 1\).
  2. Show that $$\sum _ { r = n + 1 } ^ { 2 n } ( r + 1 ) 2 ^ { r - 1 } = n 2 ^ { n } \left( 2 ^ { n + 1 } - 1 \right)$$
AQA FP2 Q5
5 The complex number \(z\) satisfies the relation $$| z + 4 - 4 i | = 4$$
  1. Sketch, on an Argand diagram, the locus of \(z\).
  2. Show that the greatest value of \(| z |\) is \(4 ( \sqrt { 2 } + 1 )\).
  3. Find the value of \(z\) for which $$\arg ( z + 4 - 4 \mathrm { i } ) = \frac { 1 } { 6 } \pi$$ Give your answer in the form \(a + \mathrm { i } b\).
AQA FP2 Q6
6 It is given that \(z = \mathrm { e } ^ { \mathrm { i } \theta }\).
    1. Show that $$z + \frac { 1 } { z } = 2 \cos \theta$$ (2 marks)
    2. Find a similar expression for $$z ^ { 2 } + \frac { 1 } { z ^ { 2 } }$$ (2 marks)
    3. Hence show that $$z ^ { 2 } - z + 2 - \frac { 1 } { z } + \frac { 1 } { z ^ { 2 } } = 4 \cos ^ { 2 } \theta - 2 \cos \theta$$ (3 marks)
  2. Hence solve the quartic equation $$z ^ { 4 } - z ^ { 3 } + 2 z ^ { 2 } - z + 1 = 0$$ giving the roots in the form \(a + \mathrm { i } b\).
AQA FP2 Q7
7
  1. Use the definitions $$\sinh \theta = \frac { 1 } { 2 } \left( \mathrm { e } ^ { \theta } - \mathrm { e } ^ { - \theta } \right) \quad \text { and } \quad \cosh \theta = \frac { 1 } { 2 } \left( \mathrm { e } ^ { \theta } + \mathrm { e } ^ { - \theta } \right)$$ to show that:
    1. \(2 \sinh \theta \cosh \theta = \sinh 2 \theta\);
    2. \(\cosh ^ { 2 } \theta + \sinh ^ { 2 } \theta = \cosh 2 \theta\).
  2. A curve is given parametrically by $$x = \cosh ^ { 3 } \theta , \quad y = \sinh ^ { 3 } \theta$$
    1. Show that $$\left( \frac { \mathrm { d } x } { \mathrm {~d} \theta } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} \theta } \right) ^ { 2 } = \frac { 9 } { 4 } \sinh ^ { 2 } 2 \theta \cosh 2 \theta$$
    2. Show that the length of the arc of the curve from the point where \(\theta = 0\) to the point where \(\theta = 1\) is $$\frac { 1 } { 2 } \left[ ( \cosh 2 ) ^ { \frac { 3 } { 2 } } - 1 \right]$$
AQA FP2 Q8
8
    1. Given that \(z ^ { 6 } - 4 z ^ { 3 } + 8 = 0\), show that \(z ^ { 3 } = 2 \pm 2 \mathrm { i }\).
    2. Hence solve the equation $$z ^ { 6 } - 4 z ^ { 3 } + 8 = 0$$ giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
  2. Show that, for any real values of \(k\) and \(\theta\), $$\left( z - k \mathrm { e } ^ { \mathrm { i } \theta } \right) \left( z - k \mathrm { e } ^ { - \mathrm { i } \theta } \right) = z ^ { 2 } - 2 k z \cos \theta + k ^ { 2 }$$
  3. Express \(z ^ { 6 } - 4 z ^ { 3 } + 8\) as the product of three quadratic factors with real coefficients.
AQA S1 Q1
1 At a certain small restaurant, the waiting time is defined as the time between sitting down at a table and a waiter first arriving at the table. This waiting time is dependent upon the number of other customers already seated in the restaurant. Alex is a customer who visited the restaurant on 10 separate days. The table shows, for each of these days, the number, \(x\), of customers already seated and his waiting time, \(y\) minutes.
\(\boldsymbol { x }\)9341081271126
\(\boldsymbol { y }\)11651191391247
  1. Calculate the equation of the least squares regression line of \(y\) on \(x\) in the form \(y = a + b x\).
  2. Give an interpretation, in context, for each of your values of \(a\) and \(b\).
  3. Use your regression equation to estimate Alex's waiting time when the number of customers already seated in the restaurant is:
    1. 5 ;
    2. 25 .
  4. Comment on the likely reliability of each of your estimates in part (c), given that, for the regression line calculated in part (a), the values of the 10 residuals lie between + 1.1 minutes and - 1.1 minutes.
AQA S1 Q2
2 Xavier, Yuri and Zara attend a sports centre for their judo club's practice sessions. The probabilities of them arriving late are, independently, \(0.3,0.4\) and 0.2 respectively.
  1. Calculate the probability that for a particular practice session:
    1. all three arrive late;
    2. none of the three arrives late;
    3. only Zara arrives late.
  2. Zara's friend, Wei, also attends the club's practice sessions. The probability that Wei arrives late is 0.9 when Zara arrives late, and is 0.25 when Zara does not arrive late. Calculate the probability that for a particular practice session:
    1. both Zara and Wei arrive late;
    2. either Zara or Wei, but not both, arrives late.
AQA S1 Q3
3 When an alarm is raised at a market town's fire station, the fire engine cannot leave until at least five fire-fighters arrive at the station. The call-out time, \(X\) minutes, is the time between an alarm being raised and the fire engine leaving the station. The value of \(X\) was recorded on a random sample of 50 occasions. The results are summarised below, where \(\bar { x }\) denotes the sample mean. $$\sum x = 286.5 \quad \sum ( x - \bar { x } ) ^ { 2 } = 45.16$$
  1. Find values for the mean and standard deviation of this sample of 50 call-out times.
  2. Hence construct a \(99 \%\) confidence interval for the mean call-out time.
  3. The fire and rescue service claims that the station's mean call-out time is less than 5 minutes, whereas a parish councillor suggests that it is more than \(6 \frac { 1 } { 2 }\) minutes. Comment on each of these claims.
AQA S1 Q4
4 The time, \(x\) seconds, spent by each of a random sample of 100 customers at an automatic teller machine (ATM) is recorded. The times are summarised in the table.
Time (seconds)Number of customers
\(20 < x \leqslant 30\)2
\(30 < x \leqslant 40\)7
\(40 < x \leqslant 60\)18
\(60 < x \leqslant 80\)27
\(80 < x \leqslant 100\)23
\(100 < x \leqslant 120\)13
\(120 < x \leqslant 150\)7
\(150 < x \leqslant 180\)3
Total100
  1. Calculate estimates for the mean and standard deviation of the time spent at the ATM by a customer.
  2. The mean time spent at the ATM by a random sample of \(\mathbf { 3 6 }\) customers is denoted by \(\bar { Y }\).
    1. State why the distribution of \(\bar { Y }\) is approximately normal.
    2. Write down estimated values for the mean and standard error of \(\bar { Y }\).
    3. Hence estimate the probability that \(\bar { Y }\) is less than \(1 \frac { 1 } { 2 }\) minutes.
AQA S1 Q5
5 [Figure 1, printed on the insert, is provided for use in this question.]
The table shows the times, in seconds, taken by a random sample of 10 boys from a junior swimming club to swim 50 metres freestyle and 50 metres backstroke.
BoyABCDEFGHIJ
Freestyle ( \(\boldsymbol { x }\) seconds)30.232.825.131.831.235.632.438.036.134.1
Backstroke ( \(y\) seconds)33.535.437.427.234.738.237.741.442.338.4
  1. On Figure 1, complete the scatter diagram for these data.
  2. Hence:
    1. give two distinct comments on what your scatter diagram reveals;
    2. state, without calculation, which of the following 3 values is most likely to be the value of the product moment correlation coefficient for the data in your scatter diagram. $$0.912 \quad 0.088 \quad 0.462$$
  3. In the sample of 10 boys, one boy is a junior-champion freestyle swimmer and one boy is a junior-champion backstroke swimmer. Identify the two most likely boys.
  4. Removing the data for the two boys whom you identified in part (c):
    1. calculate the value of the product moment correlation coefficient for the remaining 8 pairs of values of \(x\) and \(y\);
    2. comment, in context, on the value that you obtain.
AQA S1 Q6
6 Plastic clothes pegs are made in various colours.
The number of red pegs may be modelled by a binomial distribution with parameter \(p\) equal to 0.2 . The contents of packets of 50 pegs of mixed colours may be considered to be random samples.
  1. Determine the probability that a packet contains:
    1. less than or equal to 15 red pegs;
    2. exactly 10 red pegs;
    3. more than 5 but fewer than 15 red pegs.
  2. Sly, a student, claims to have counted the number of red pegs in each of 100 packets of 50 pegs. From his results the following values are calculated. Mean number of red pegs per packet \(= 10.5\) Variance of number of red pegs per packet \(= 20.41\) Comment on the validity of Sly's claim.
AQA S1 Q7
7
  1. The weight, \(X\) grams, of soup in a carton may be modelled by a normal random variable with mean 406 and standard deviation 4.2. Find the probability that the weight of soup in a carton:
    1. is less than 400 grams;
    2. is between 402.5 grams and 407.5 grams.
  2. The weight, \(Y\) grams, of chopped tomatoes in a tin is a normal random variable with mean \(\mu\) and standard deviation \(\sigma\).
    1. Given that \(\mathrm { P } ( Y < 310 ) = 0.975\), explain why: $$310 - \mu = 1.96 \sigma$$
    2. Given that \(\mathrm { P } ( Y < 307.5 ) = 0.86\), find, to two decimal places, values for \(\mu\) and \(\sigma\).
      (4 marks)