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AQA Further Paper 2 2024 June Q4
1 marks Easy -1.2
4 The function f is a quartic function with real coefficients.
The complex number 5i is a root of the equation \(\mathrm { f } ( x ) = 0\) Which one of the following must be a factor of \(\mathrm { f } ( x )\) ?
Circle your answer.
( \(x ^ { 2 } - 25\) ) \(\left( x ^ { 2 } - 5 \right)\) \(\left( x ^ { 2 } + 5 \right)\) \(\left( x ^ { 2 } + 25 \right)\)
AQA Further Paper 2 2024 June Q5
3 marks Moderate -0.5
5 The first four terms of the series \(S\) can be written as $$S = ( 1 \times 2 ) + ( 2 \times 3 ) + ( 3 \times 4 ) + ( 4 \times 5 ) + \ldots$$ 5
  1. Write an expression, using \(\sum\) notation, for the sum of the first \(n\) terms of \(S\) 5
  2. Show that the sum of the first \(n\) terms of \(S\) is equal to $$\frac { 1 } { 3 } n ( n + 1 ) ( n + 2 )$$
AQA Further Paper 2 2024 June Q6
3 marks Moderate -0.5
6 The cubic equation $$x ^ { 3 } + 5 x ^ { 2 } - 4 x + 2 = 0$$ has roots \(\alpha , \beta\) and \(\gamma\) Find a cubic equation, with integer coefficients, whose roots are \(3 \alpha , 3 \beta\) and \(3 \gamma\)
AQA Further Paper 2 2024 June Q8
4 marks Standard +0.8
8 The vectors \(\mathbf { a } , \mathbf { b }\), and \(\mathbf { c }\) are such that \(\mathbf { a } \times \mathbf { b } = \left[ \begin{array} { l } 2 \\ 1 \\ 0 \end{array} \right]\) and \(\mathbf { a } \times \mathbf { c } = \left[ \begin{array} { l } 0 \\ 0 \\ 3 \end{array} \right]\) Work out \(( \mathbf { a } - \mathbf { 4 } \mathbf { b } + \mathbf { 3 c } ) \times ( \mathbf { 2 a } )\) [0pt] [4 marks]
AQA Further Paper 2 2024 June Q9
4 marks Challenging +1.2
9 A curve passes through the point (-2, 4.73) and satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y ^ { 2 } - x ^ { 2 } } { 2 x + 3 y }$$ 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 0.02 , to estimate the value of \(y\) when \(x = - 1.96\) Give your answer to five significant figures.
[0pt] [4 marks]
AQA Further Paper 2 2024 June Q10
4 marks Standard +0.8
10 The matrix \(\mathbf { C }\) is defined by $$\mathbf { C } = \left[ \begin{array} { c c } 3 & 2 \\ - 4 & 5 \end{array} \right]$$ Prove that the transformation represented by \(\mathbf { C }\) has no invariant lines of the form \(y = k x\) Latifa and Sam are studying polynomial equations of degree greater than 2 , with real coefficients and no repeated roots. Latifa says that if such an equation has exactly one real root, it must be of degree 3 Sam says that this is not correct. State, giving reasons, whether Latifa or Sam is right.
AQA Further Paper 2 2024 June Q12
5 marks Challenging +1.2
12
The transformation S is represented by the matrix \(\mathbf { M } = \left[ \begin{array} { c c } 1 & - 6 \\ 2 & 7 \end{array} \right]\) The transformation T is a reflection in the line \(y = x \sqrt { 3 }\) and is represented by the matrix \(\mathbf { N }\) The point \(P ( x , y )\) is transformed first by S , then by T
The result of these transformations is the point \(Q ( 3,8 )\) Find the coordinates of \(P\) Give your answers to three decimal places.
AQA Further Paper 2 2024 June Q13
8 marks Standard +0.8
13
  1. Use the method of differences to show that $$\sum _ { r = 2 } ^ { n } \frac { 1 } { ( r - 1 ) r ( r + 1 ) } = \frac { 1 } { 4 } - \frac { 1 } { 2 n } + \frac { 1 } { 2 ( n + 1 ) }$$ [5 marks]
    13
  2. Find the smallest integer \(n\) such that $$\sum _ { r = 2 } ^ { n } \frac { 1 } { ( r - 1 ) r ( r + 1 ) } > 0.24999$$
AQA Further Paper 2 2024 June Q14
10 marks Challenging +1.2
14 The matrix \(\mathbf { M }\) is defined as $$\mathbf { M } = \left[ \begin{array} { c c c } 5 & 2 & 1 \\ 6 & 3 & 2 k + 3 \\ 2 & 1 & 5 \end{array} \right]$$ where \(k\) is a constant. 14
  1. Given that \(\mathbf { M }\) is a non-singular matrix, find \(\mathbf { M } ^ { - 1 }\) in terms of \(k\) 14
  2. State any restrictions on the value of \(k\) 14
  3. Using your answer to part (a), show that the solution to the set of simultaneous equations below is independent of the value of \(k\) $$\begin{array} { r l c c } 5 x + 2 y + c & = & 1 \\ 6 x + 3 y + ( 2 k + 3 ) z & = & 4 k + 3 \\ 2 x + y + 5 z & = & 9 \end{array}$$
AQA Further Paper 2 2024 June Q15
7 marks Standard +0.8
15 The diagram shows the line \(y = 5 - x\) \includegraphics[max width=\textwidth, alt={}, center]{99b03f18-6dd6-437d-8b01-009ca7ab49ea-18_1255_1125_349_440} 15
  1. On the diagram above, sketch the graph of \(y = \left| x ^ { 2 } - 4 x \right|\), including all parts of the graph where it intersects the line \(y = 5 - x\) (You do not need to show the coordinates of the points of intersection.) 15
  2. Find the solution of the inequality $$\left| x ^ { 2 } - 4 x \right| > 5 - x$$ Give your answer in an exact form.
    [0pt] [4 marks]
AQA Further Paper 2 2024 June Q16
9 marks Challenging +1.2
16 The function f is defined by $$f ( x ) = \frac { a x + 5 } { x + b }$$ where \(a\) and \(b\) are constants. The graph of \(y = \mathrm { f } ( x )\) has asymptotes \(x = - 2\) and \(y = 3\) 16
  1. Write down the value of \(a\) and the value of \(b\) 16
  2. The diagram shows the graph of \(y = \mathrm { f } ( x )\) and its asymptotes.
    The shaded region \(R\) is enclosed by the graph of \(y = \mathrm { f } ( x )\), the \(x\)-axis and the \(y\)-axis. \includegraphics[max width=\textwidth, alt={}, center]{99b03f18-6dd6-437d-8b01-009ca7ab49ea-20_858_1002_1267_504} 16
    1. The shaded region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis to form a solid. Find the volume of this solid. Give your answer to three significant figures. 16
  4. (ii) The shaded region \(R\) is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis to form a solid.
    Find the volume of this solid.
    Give your answer to three significant figures.
    [0pt] [4 marks]
AQA Further Paper 2 2024 June Q17
9 marks Challenging +1.2
17 The Argand diagram below shows a circle \(C\) \includegraphics[max width=\textwidth, alt={}, center]{99b03f18-6dd6-437d-8b01-009ca7ab49ea-22_1063_926_317_541} 17
  1. Write down the equation of the locus of \(C\) in the form $$| z - w | = a$$ where \(w\) is a complex number whose real and imaginary parts are integers, and \(a\) is an integer.
    17
  2. It is given that \(z _ { 1 }\) is a complex number representing a point on \(C\). Of all the complex numbers which represent points on \(C , z _ { 1 }\) has the least argument. 17
    1. Find \(\left| z _ { 1 } \right|\) Give your answer in an exact form.
      17
  4. (ii) Show that \(\arg z _ { 1 } = \arcsin \left( \frac { 6 \sqrt { 3 } - 2 } { 13 } \right)\)
    \includegraphics[max width=\textwidth, alt={}]{99b03f18-6dd6-437d-8b01-009ca7ab49ea-25_2486_1744_178_132}
AQA Further Paper 2 2024 June Q19
10 marks Challenging +1.2
19 Solve the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } - 45 y = 21 \mathrm { e } ^ { 5 x } - 0.3 x + 27 x ^ { 2 }$$ given that \(y = \frac { 37 } { 225 }\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) when \(x = 0\) [0pt] [10 marks]
  • \(\begin{gathered} \text { - } \\ \text { - } \\ \text { - } \\ \text { - } \\ \text { - } \\ \text { - } \\ \text { - } \end{gathered}\)
AQA Further Paper 2 2024 June Q20
9 marks Challenging +1.2
20 The integral \(I _ { n }\) is defined by $$I _ { n } = \int _ { 0 } ^ { \frac { \pi } { 4 } } \cos ^ { n } x \mathrm {~d} x \quad ( n \geq 0 )$$ 20
  1. Show that $$I _ { n } = \left( \frac { n - 1 } { n } \right) I _ { n - 2 } + \frac { 1 } { n \left( 2 ^ { \frac { n } { 2 } } \right) } \quad ( n \geq 2 )$$ 20
  2. Use the result from part (a) to show that $$\int _ { 0 } ^ { \frac { \pi } { 4 } } \cos ^ { 6 } x d x = \frac { a \pi + b } { 192 }$$ where \(a\) and \(b\) are integers to be found. \includegraphics[max width=\textwidth, alt={}]{99b03f18-6dd6-437d-8b01-009ca7ab49ea-31_2491_1755_173_123} number \section*{Additional page, if required.
    Additional page, if required. Write the question numbers in the left-hand margin.} Additional page, if required. number Additional page, if required.
    Write the question numbers in the left-hand margin. number \section*{Additional page, if required.
    Additional page, if required. Write the question numbers in the left-hand margin.} number\section*{Additional page, if required.
    Additional page, if required. Write the question numbers in the left-hand margin.}
    \includegraphics[max width=\textwidth, alt={}]{99b03f18-6dd6-437d-8b01-009ca7ab49ea-36_2487_1748_175_130}
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
    1. State a criticism of Roy's model. 4
  4. (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
  3. (ii) the office receives more than 30 calls in an hour.
    8
  4. 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
  5. The office has just received a call.
    8
    1. Find the probability that the next call is received more than 10 minutes later.
      8
  7. (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.