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AQA Further Paper 2 2022 June Q6
3 marks Challenging +1.2
6 The diagram below shows part of the graph of \(y = \mathrm { f } ( x )\) The line \(T P Q\) is a tangent to the graph of \(y = \mathrm { f } ( x )\) at the point \(P \left( \frac { a + b } { 2 } , \mathrm { f } \left( \frac { a + b } { 2 } \right) \right)\) The points \(S ( a , 0 )\) and \(T\) lie on the line \(x = a\) The points \(Q\) and \(R ( b , 0 )\) lie on the line \(x = b\) \includegraphics[max width=\textwidth, alt={}, center]{74b8239a-1f46-45e7-ad20-2dce7bf4baf6-05_748_696_669_671} Sharon uses the mid-ordinate rule with one strip to estimate the value of the integral \(\int _ { a } ^ { b } \mathrm { f } ( x ) \mathrm { d } x\) By considering the area of the trapezium QRST, state, giving reasons, whether you would expect Sharon's estimate to be an under-estimate or an over-estimate.
AQA Further Paper 2 2022 June Q7
8 marks Standard +0.8
7 The function f is defined by $$\mathrm { f } ( x ) = \frac { a x - 5 } { 2 x + b } \quad x \in \mathbb { R } , x \neq \frac { 9 } { 2 }$$ where \(a\) and \(b\) are integers.
The graph of \(y = \mathrm { f } ( x )\) has asymptotes \(x = \frac { 9 } { 2 }\) and \(y = 3\) 7
  1. Find the value of \(a\) and the value of \(b\) 7
  2. Solve the inequality $$\mathrm { f } ( x ) \leq x + 2$$ Fully justify your answer.
AQA Further Paper 2 2022 June Q8
10 marks Challenging +1.2
8
  1. The function f is defined as \(\mathrm { f } ( x ) = \sec x\) 8
    1. (i) Show that \(\mathrm { f } ^ { ( 4 ) } ( 0 ) = 5\) 8
    2. (ii) Hence find the first three non-zero terms of the Maclaurin series for \(\mathrm { f } ( x ) = \sec x\) 8
    3. Prove that $$\lim _ { x \rightarrow 0 } \left( \frac { \sec x - \cosh x } { x ^ { 4 } } \right) = \frac { 1 } { 6 }$$
AQA Further Paper 2 2022 June Q9
14 marks Standard +0.8
9
  1. A curve passes through the point (5, 12.3) and satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \left( x ^ { 2 } - 9 \right) ^ { \frac { 1 } { 2 } } + \frac { 2 x y } { x ^ { 2 } - 9 } \quad x > 3$$ Use Euler's step by step method once, and then the midpoint formula $$y _ { r + 1 } = y _ { r - 1 } + 2 h \mathrm { f } \left( x _ { r } , y _ { r } \right) , \quad x _ { r + 1 } = x _ { r } + h$$ once, each with a step length of 0.1 , to estimate the value of \(y\) when \(x = 5.2\) Give your answer to six significant figures.
    9
  2. (i) Find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \left( x ^ { 2 } - 9 \right) ^ { \frac { 1 } { 2 } } + \frac { 2 x y } { x ^ { 2 } - 9 } \quad ( x > 3 )$$ 9 (b) (ii) Given that \(y\) satisfies the differential equation in part (b)(i) and that \(y = 12.3\) when \(x = 5\), find the value of \(y\) when \(x = 5.2\) Give your answer to six significant figures.
    [0pt] [3 marks]
    9
  3. Comment on the accuracy of your answer to part (a).
AQA Further Paper 2 2022 June Q10
7 marks Challenging +1.2
10 The curve \(C _ { 1 }\) has equation $$\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 4 } = 1$$ The curve \(C _ { 2 }\) has equation $$x ^ { 2 } - 25 y ^ { 2 } - 6 x - 200 y - 416 = 0$$ 10
  1. Find a sequence of transformations that maps the graph of \(C _ { 1 }\) onto the graph of \(C _ { 2 }\) [4 marks]
    10
  2. Find the equations of the asymptotes to \(C _ { 2 }\) Give your answers in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
AQA Further Paper 2 2022 June Q11
9 marks Standard +0.3
11
  1. Find the eigenvalues and corresponding eigenvectors of the matrix $$\mathbf { M } = \left[ \begin{array} { c c } \frac { 5 } { 2 } & - \frac { 3 } { 2 } \\ - \frac { 3 } { 2 } & \frac { 13 } { 2 } \end{array} \right]$$ 11
  2. (i) Describe how the directions of the invariant lines of the transformation represented by \(\mathbf { M }\) are related to each other. Fully justify your answer.
    [0pt] [2 marks]
    11 (b) (ii) Describe fully the transformation represented by \(\mathbf { M }\)
AQA Further Paper 2 2022 June Q12
11 marks Standard +0.3
12 The shaded region shown in the diagram below is bounded by the \(x\)-axis, the curve \(y = \mathrm { f } ( x )\), and the lines \(x = a\) and \(x = b\) \includegraphics[max width=\textwidth, alt={}, center]{74b8239a-1f46-45e7-ad20-2dce7bf4baf6-16_661_721_406_662} The shaded region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid.
12
  1. Show that the volume of this solid is $$\pi \int _ { a } ^ { b } ( \mathrm { f } ( x ) ) ^ { 2 } \mathrm {~d} x$$ 12
  2. In the case where \(a = 1 , b = 2\) and $$f ( x ) = \frac { x + 3 } { ( x + 1 ) \sqrt { x } }$$ show that the volume of the solid is $$\pi \left( \ln \left( \frac { 2 ^ { m } } { 3 ^ { n } } \right) - \frac { 2 } { 3 } \right)$$ where \(m\) and \(n\) are integers.
AQA Further Paper 2 2022 June Q13
16 marks Challenging +1.2
13
  1. The matrix A represents a reflection in the line \(y = m x\), where \(m\) is a constant. Show that \(\mathbf { A } = \left( \frac { 1 } { m ^ { 2 } + 1 } \right) \left[ \begin{array} { c c } 1 - m ^ { 2 } & 2 m \\ 2 m & m ^ { 2 } - 1 \end{array} \right]\) You may use the result in the formulae booklet. 13
  2. \(\quad\) The matrix \(\mathbf { B }\) is defined as \(\mathbf { B } = \left[ \begin{array} { l l } 3 & 0 \\ 0 & 3 \end{array} \right]\) Show that \(( \mathbf { B A } ) ^ { 2 } = k \mathbf { I }\) where \(\mathbf { I }\) is the \(2 \times 2\) identity matrix and \(k\) is an integer.
    13
  3. (i) The diagram below shows a point \(P\) and the line \(y = m x\) Draw four lines on the diagram to demonstrate the result proved in part (b).
    Label as \(P ^ { \prime }\) the image of \(P\) under the transformation represented by (BA) \({ } ^ { 2 }\) \includegraphics[max width=\textwidth, alt={}, center]{74b8239a-1f46-45e7-ad20-2dce7bf4baf6-20_579_1068_584_488} 13 (c) (ii) Explain how your completed diagram shows the result proved in part (b).
    13
  4. The matrix \(\mathbf { C }\) is defined as \(\mathbf { C } = \left[ \begin{array} { c c } \frac { 12 } { 5 } & \frac { 9 } { 5 } \\ \frac { 9 } { 5 } & - \frac { 12 } { 5 } \end{array} \right]\) Find the value of \(m\) such that \(\mathbf { C } = \mathbf { B A }\) Fully justify your answer.
    [0pt] [4 marks]
AQA Further Paper 2 2022 June Q14
14 marks Challenging +1.8
14 On an isolated island some rabbits have been accidently introduced. In order to eliminate them, conservationists have introduced some birds of prey.
At time \(t\) years \(( t \geq 0 )\) there are \(x\) rabbits and \(y\) birds of prey.
At time \(t = 0\) there are 1755 rabbits and 30 birds of prey.
When \(t > 0\) it is assumed that:
  • the rabbits will reproduce at a rate of \(a \%\) per year
  • each bird of prey will kill, on average, \(b\) rabbits per year
  • the death rate of the birds of prey is \(c\) birds per year
  • the number of birds of prey will increase at a rate of \(d \%\) of the rabbit population per year.
This system is represented by the coupled differential equations: $$\begin{aligned} & \frac { \mathrm { d } x } { \mathrm {~d} t } = 0.4 x - 13 y \\ & \frac { \mathrm {~d} y } { \mathrm {~d} t } = 0.01 x - 1.95 \end{aligned}$$ 14
  1. State the value of \(a\), the value of \(b\), the value of \(c\) and the value of \(d\) [0pt] [2 marks]
    14
  2. Solve the coupled differential equations to find both \(x\) and \(y\) in terms of \(t\)
AQA Further Paper 3 Statistics 2019 June Q1
1 marks Easy -1.2
1 The discrete random variable \(X\) has \(\operatorname { Var } ( X ) = 5\) Find \(\operatorname { Var } ( 4 X - 3 )\) Circle your answer.
17207780
AQA Further Paper 3 Statistics 2019 June Q2
1 marks Standard +0.8
2 Amy takes a sample of size 50 from a normal distribution with mean \(\mu\) and variance 16 She conducts a hypothesis test with hypotheses: $$\begin{aligned} & \mathrm { H } _ { 0 } : \mu = 52 \\ & \mathrm { H } _ { 1 } : \mu > 52 \end{aligned}$$ She rejects the null hypothesis if her sample has a mean greater than 53
The actual population mean is 53.5
Find the probability that Amy makes a Type II error.
Circle your answer. \(0.4 \% 3.9 \% 18.9 \% 15.0 \%\)
AQA Further Paper 3 Statistics 2019 June Q3
4 marks Standard +0.8
3 Alan's journey time to work can be modelled by a normal distribution with standard deviation 6 minutes. Alan measures the journey time to work for a random sample of 5 journeys. The mean of the 5 journey times is 36 minutes. 3
  1. Construct a 95\% confidence interval for Alan's mean journey time to work, giving your values to one decimal place.
    3
  2. Alan claims that his mean journey time to work is 30 minutes.
    State, with a reason, whether or not the confidence interval found in part (a) supports Alan's claim.
    3
  3. Suppose that the standard deviation is not known but a sample standard deviation is found from Alan's sample and calculated to be 6 Explain how the working in part (a) would change.
AQA Further Paper 3 Statistics 2019 June Q4
7 marks Standard +0.3
4 A random variable \(X\) has a rectangular distribution. The mean of \(X\) is 3 and the variance of \(X\) is 3
4
  1. Determine the probability density function of \(X\).
    Fully justify your answer. 4
  2. A 6 metre clothes line is connected between the point \(P\) on one building and the point \(Q\) on a second building. Roy is concerned the clothes line may break. He uses the random variable \(X\) to model the distance in metres from \(P\) where the clothes line breaks. 4 (b) (i) State a criticism of Roy's model. 4 (b) (ii) On the axes below, sketch the probability density function for an alternative model for the clothes line. \includegraphics[max width=\textwidth, alt={}, center]{3219e2fe-7757-469a-9d0d-654b3e180e8d-05_584_1162_1210_438}
AQA Further Paper 3 Statistics 2019 June Q5
7 marks Standard +0.3
5 An insurance company models the claims it pays out in pounds \(( \pounds )\) with a random variable \(X\) which has probability density function $$f ( x ) = \begin{cases} \frac { k } { x } & 1 < x < a \\ 0 & \text { otherwise } \end{cases}$$ 5
  1. The median claim is \(\pounds 200\) Show that \(k = \frac { 1 } { 2 \ln 200 }\) 5
  2. Find \(\mathrm { P } ( X < 2000 )\), giving your answer to three significant figures.
    5
  3. The insurance company finds that the maximum possible claim is \(\pounds 2000\) and they decide to refine their probability density function. Suggest how this could be done.
AQA Further Paper 3 Statistics 2019 June Q6
9 marks Standard +0.3
6 During August, 102 candidates took their driving test at centre \(A\) and 60 passed. During the same month, 110 candidates took their driving test at centre \(B\) and 80 passed. 6
  1. Test whether the driving test result is independent of the driving test centre using the \(5 \%\) level of significance. 6
  2. Rebecca claims that if the result of the test in part (a) is to reject the null hypothesis then it is easier to pass a driving test at centre \(B\) than centre \(A\). State, with a reason, whether or not you agree with Rebecca's claim.
AQA Further Paper 3 Statistics 2019 June Q7
9 marks Standard +0.3
7 A shopkeeper sells chocolate bars which are described by the manufacturer as having an average mass of 45 grams. The shopkeeper claims that the mass of the chocolate bars, \(X\) grams, is getting smaller on average. A random sample of 6 chocolate bars is taken and their masses in grams are measured. The results are $$\sum x = 246 \quad \text { and } \quad \sum x ^ { 2 } = 10198$$ Investigate the shopkeeper's claim using the \(5 \%\) level of significance.
State any assumptions that you make.
AQA Further Paper 3 Statistics 2019 June Q8
12 marks Standard +0.3
8 The number of telephone calls received by an office can be modelled by a Poisson distribution with mean 3 calls per 10 minutes. 8
  1. Find the probability that:
    8
      1. the office receives exactly 2 calls in 10 minutes; 8
      2. the office receives more than 30 calls in an hour.
        8
      3. The office manager splits an hour into 6 periods of 10 minutes and records the number of telephone calls received in each of the 10 minute periods. Find the probability that the office receives exactly 2 calls in a 10 minute period exactly twice within an hour.
        8
      4. The office has just received a call.
      8
      1. Find the probability that the next call is received more than 10 minutes later.
        8
    3. (ii) Mahah arrives at the office 5 minutes after the last call was received.
      State the probability that the next call received by the office is received more than 10 minutes later. Explain your answer. \includegraphics[max width=\textwidth, alt={}, center]{3219e2fe-7757-469a-9d0d-654b3e180e8d-14_2492_1721_217_150} Additional page, if required.
      Write the question numbers in the left-hand margin. Additional page, if required.
      Write the question numbers in the left-hand margin.
AQA Further Paper 3 Statistics 2020 June Q1
1 marks Easy -1.2
1 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 1 } { 5 } & 1 \leq x \leq 6 \\ 0 & \text { otherwise } \end{cases}$$ Find \(\mathrm { P } ( X \geq 3 )\) Circle your answer. \(\frac { 1 } { 5 } \quad \frac { 2 } { 5 } \quad \frac { 3 } { 5 } \quad \frac { 4 } { 5 }\)
AQA Further Paper 3 Statistics 2020 June Q2
1 marks Easy -1.2
2 Jamie is conducting a hypothesis test on a random variable which has a normal distribution with standard deviation 1 The hypotheses are $$\begin{aligned} & \mathrm { H } _ { 0 } : \mu = 5 \\ & \mathrm { H } _ { 1 } : \mu > 5 \end{aligned}$$ He takes a random sample of size 4
The mean of his sample is 6
He uses a 5\% level of significance.
Before Jamie conducted the test, what was the probability that he would make a Type I error? Circle your answer.
[0pt] [1 mark] \(0.0228 \quad 0.0456 \quad 0.0500 \quad 0.1587\)
AQA Further Paper 3 Statistics 2020 June Q3
4 marks Standard +0.3
3 The mass of male giraffes is assumed to have a normal distribution. Duncan takes a random sample of 600 male giraffes.
The mean mass of the sample is 1196 kilograms.
The standard deviation of the sample is 98 kilograms.
3
  1. Construct a 94\% confidence interval for the mean mass of male giraffes, giving your values to one decimal place.
    3
  2. Explain whether or not your answer to part (a) would change if a sample of size 5 was taken with the same mean and standard deviation.
AQA Further Paper 3 Statistics 2020 June Q4
9 marks Standard +0.3
4 The discrete random variable \(X\) follows a discrete uniform distribution and takes values \(1,2,3 , \ldots , n\). The discrete random variable \(Y\) is defined by \(Y = 2 X\) 4
  1. Using the standard results for \(\sum n , \sum n ^ { 2 }\) and \(\operatorname { Var } ( a X + b )\), prove that $$\operatorname { Var } ( Y ) = \frac { n ^ { 2 } - 1 } { 3 }$$ 4
  2. A spinning toy can land on one of four values: 2, 4, 6 or 8
    Using a discrete uniform distribution, find the probability that the next value the toy lands on is greater than 2 4
  3. State an assumption that is required for the discrete uniform distribution used in part (b) to be valid.
AQA Further Paper 3 Statistics 2020 June Q5
7 marks Challenging +1.2
5 Emily claims that the average number of runners per minute passing a shop during a long distance run is 8 Emily conducts a hypothesis test to investigate her claim.
5
  1. State the hypotheses for Emily's test. 5
  2. Emily counts the number of runners, \(X\), passing the shop in a randomly chosen minute. The critical region for Emily's test is \(X \leq 2\) or \(X \geq 14\) During a randomly chosen minute, Emily counts 3 runners passing the shop.
    Determine the outcome of Emily's hypothesis test.
    5
  3. The actual average number of runners per minute passing the shop is 7 Find the power of Emily's hypothesis test, giving your answer to three significant figures.
AQA Further Paper 3 Statistics 2020 June Q6
8 marks Standard +0.3
6 The distance, \(X\) metres, between successive breaks in a water pipe is modelled by an exponential distribution. The mean of \(X\) is 25 The distance between two successive breaks is measured. A water pipe is given a 'Red' rating if the distance is less than \(d\) metres. The government has introduced a new law changing \(d\) to 2
Before the government introduced the new law, the probability that a water pipe is given a 'Red' rating was 0.05 6
  1. Explain whether or not the probability that a water pipe is given a 'Red' rating has increased as a result of the new law.
    6
  2. Find the probability density function of the random variable \(X\). 6
  3. After investigation, the distances between successive breaks in water pipes are found to have a standard deviation of 5 metres. Explain whether or not the use of an exponential model in parts (a) and (b) is appropriate.
    [0pt] [2 marks]
AQA Further Paper 3 Statistics 2020 June Q7
8 marks Standard +0.3
7 The rainfall per day in February in a particular town has been recorded as having a mean of 1.8 inches. Sienna claims that rainfall in February has increased in the town. She records the rainfall in a random sample of 12 days. Her sample mean is 2 inches and her sample standard deviation is 0.4 inches.
It is assumed that rainfall per day has a normal distribution.
7
  1. Investigate Sienna's claim using the \(5 \%\) level of significance.
    7
  2. For the test carried out in part (a), state in context the meaning of a Type II error. 7
  3. The distribution of rainfall per day in February in the town over 10 years is shown in the histogram. \includegraphics[max width=\textwidth, alt={}, center]{443e7f17-a555-41ff-9d91-541cf45aae99-11_508_645_849_699} Explain whether or not the assumption that rainfall per day in February has a normal distribution is appropriate.
AQA Further Paper 3 Statistics 2020 June Q8
6 marks Standard +0.3
8 Ray is conducting a hypothesis test with the hypotheses \(\mathrm { H } _ { 0 }\) : There is no association between time of day and number of snacks eaten \(\mathrm { H } _ { 1 }\) : There is an association between time of day and number of snacks eaten
He calculates expected frequencies correct to two decimal places, which are given in the following table.
Number of snacks eaten
\cline { 2 - 5 }\cline { 2 - 4 }012 or more
\cline { 2 - 4 } Time of Day23.6821.055.26
\cline { 2 - 5 }Night21.3218.954.74
\cline { 2 - 5 }
\cline { 2 - 5 }
Ray calculates his test statistic using \(\sum \frac { ( O - E ) ^ { 2 } } { E }\) 8
  1. State, with a reason, the error Ray has made and describe any changes Ray will need to make to his test.
    8
  2. Having made the necessary corrections as described in part (a), the correct value of the test statistic is 8.74 Complete Ray's hypothesis test using a \(1 \%\) level of significance.