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AQA Further AS Paper 1 2023 June Q7
7 marks Standard +0.3
7
  1. Show that, for all integers \(r\), $$\frac { 1 } { 2 r - 1 } - \frac { 1 } { 2 r + 1 } = \frac { 2 } { ( 2 r - 1 ) ( 2 r + 1 ) }$$ 7
  2. Hence, using the method of differences, show that $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) } = \frac { a n } { b n + c }$$ where \(a\), \(b\) and \(c\) are integers to be determined.
    7
  3. Hence, or otherwise, evaluate $$\frac { 1 } { 1 \times 3 } + \frac { 1 } { 3 \times 5 } + \frac { 1 } { 5 \times 7 } + \ldots + \frac { 1 } { 99 \times 101 }$$
AQA Further AS Paper 1 2023 June Q8
4 marks Moderate -0.3
8 Abdoallah wants to write the complex number \(- 1 + \mathrm { i } \sqrt { 3 }\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\) where \(r \geq 0\) and \(- \pi < \theta \leq \pi\) Here is his method: $$\begin{array} { r l r l } r & = \sqrt { ( - 1 ) ^ { 2 } + ( \sqrt { 3 } ) ^ { 2 } } & & \tan \theta = \frac { \sqrt { 3 } } { - 1 } \\ & = \sqrt { 1 + 3 } & & \Rightarrow \\ & = \sqrt { 4 } & & \tan \theta = - \sqrt { 3 } \\ & = 2 & & \theta = \tan ^ { - 1 } ( - \sqrt { 3 } ) \\ & & \theta = - \frac { \pi } { 3 } \\ & - 1 + i \sqrt { 3 } = 2 \left( \cos \left( - \frac { \pi } { 3 } \right) + i \sin \left( - \frac { \pi } { 3 } \right) \right) \end{array}$$ There is an error in Abdoallah's method. 8
  1. Show that Abdoallah's answer is wrong by writing $$2 \left( \cos \left( - \frac { \pi } { 3 } \right) + i \sin \left( - \frac { \pi } { 3 } \right) \right)$$ in the form \(x + \mathrm { i } y\) Simplify your answer.
    8
  2. Explain the error in Abdoallah's method.
    8
  3. Express \(- 1 + \mathrm { i } \sqrt { 3 }\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\) 8
  4. Write down the complex conjugate of \(- 1 + i \sqrt { 3 }\)
AQA Further AS Paper 1 2023 June Q9
11 marks Standard +0.3
9 The matrix \(\mathbf { M }\) represents the transformation T and is given by $$\mathbf { M } = \left[ \begin{array} { c c } 3 p + 1 & 12 \\ p + 2 & p ^ { 2 } - 3 \end{array} \right]$$ 9
  1. In the case when \(p = 0\) show that the image of the point \(( 4,5 )\) under T is the point \(( 64 , - 7 )\) 9
  2. In the case when \(p = - 2\) find the gradient of the line of invariant points under \(T\) 9
  3. Show that \(p = 3\) is the only real value of \(p\) for which \(\mathbf { M }\) is singular.
    The curve \(C\) has equation $$y = \frac { 3 x ^ { 2 } + m x + p } { x ^ { 2 } + p x + m }$$ where \(m\) and \(p\) are integers.
    The vertical asymptotes of \(C\) are \(x = - 4\) and \(x = - 1\) The curve \(C\) is shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{b37e2ee7-1cde-4d75-895a-381b32f4e95a-12_867_1102_733_463}
AQA Further AS Paper 1 2023 June Q10
9 marks Standard +0.3
10
  1. Write down the equation of the horizontal asymptote of \(C\) 10
  2. Find the value of \(m\) and the value of \(p\)
    10
  3. 10
  		4. Hence, or otherwise, write down the coordinates of the \(y\)-intercept of \(C\)
    Without using calculus, show that the line \(y = - 1\) does not intersect \(C\)
AQA Further AS Paper 1 2023 June Q11
8 marks Moderate -0.5
11 A point has Cartesian coordinates \(( x , y )\) and polar coordinates \(( r , \theta )\) where \(r \geq 0\) and \(- \pi < \theta \leq \pi\) 11
  1. Express \(r\) in terms of \(x\) and \(y\) 11
  2. Express \(x\) in terms of \(r\) and \(\theta\) 11
  3. The curve \(C _ { 1 }\) has the polar equation $$r ( 2 + \cos \theta ) = 1 \quad - \pi < \theta \leq \pi$$ 11 (c) (i) Show that the Cartesian equation of \(C _ { 1 }\) can be written as $$a y ^ { 2 } = ( 1 + b x ) ( 1 + x )$$ where \(a\) and \(b\) are integers to be determined.
    11 (c) (ii) The curve \(C _ { 2 }\) has the Cartesian equation $$a x ^ { 2 } = ( 1 + b y ) ( 1 + y )$$ where \(a\) and \(b\) take the same values as in part (c)(i). Describe fully a single transformation that maps the curve \(C _ { 1 }\) onto the curve \(C _ { 2 }\)
AQA Further AS Paper 1 2023 June Q12
13 marks Standard +0.3
12
  1. Show that \(( 1 + i ) ^ { 4 } = - 4\) 12
  2. The function f is defined by $$f ( z ) = z ^ { 4 } + 3 z ^ { 2 } - 6 z + 10 \quad z \in \mathbb { C }$$ 12 (b) (i) Show that (1+i) is a root of \(\mathrm { f } ( \mathrm { z } ) = 0\) 12 (b) (ii) Hence write down another root of \(\mathrm { f } ( \mathrm { z } ) = 0\) 12 (b) (iii) One of the linear factors of \(\mathrm { f } ( \mathrm { z } )\) is $$( z - ( 1 + i ) )$$ Write down another linear factor and hence, or otherwise, find a quadratic factor of \(\mathrm { f } ( \mathrm { z } )\) with real coefficients.
    12 (b) (iv) Find another quadratic factor of \(\mathrm { f } ( \mathrm { z } )\) with real coefficients.
    12 (b) (v) Hence explain why the graph of \(y = \mathrm { f } ( x )\) does not intersect the \(x\)-axis.
AQA Further AS Paper 1 2023 June Q13
10 marks Standard +0.3
13
  1. Prove by induction that, for all integers \(n \geq 1\), $$\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )$$ [4 marks]
    13
  2. Hence, or otherwise, write down a factorised expression for the sum of the first \(2 n\) squares $$1 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } + \ldots + ( 2 n ) ^ { 2 }$$ 13
  3. Use the formula in part (a) to write down a factorised expression for the sum of the first \(n\) even squares $$2 ^ { 2 } + 4 ^ { 2 } + 6 ^ { 2 } + \ldots + ( 2 n ) ^ { 2 }$$ 13
  4. Hence, or otherwise, show that the sum of the first \(n\) odd squares is $$a n ( b n - 1 ) ( b n + 1 )$$ where \(a\) and \(b\) are rational numbers to be determined.
AQA Further AS Paper 1 2023 June Q14
4 marks Standard +0.8
14 The inequality $$\left( x ^ { 2 } - 5 x - 24 \right) \left( x ^ { 2 } + 7 x + a \right) < 0$$ has the solution set $$\{ x : - 9 < x < - 3 \} \cup \{ x : 2 < x < b \}$$ Find the values of integers \(a\) and \(b\) \includegraphics[max width=\textwidth, alt={}]{b37e2ee7-1cde-4d75-895a-381b32f4e95a-21_2491_1755_173_123} number Additional page, if required. Write the question numbers in the left-hand margin. \(\_\_\_\_\) number \section*{Additional page, if required. Write the question numbers in the left-hand margin.
Additional page, if required. uestion numbers in the left-hand margin.}
Question numberAdditional page, if required. Write the question numbers in the left-hand margin.
\(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
AQA Further AS Paper 2 Statistics 2021 June Q1
1 marks Easy -1.2
1 The discrete random variable \(X\) has \(\operatorname { Var } ( X ) = 6.5\) Find \(\operatorname { Var } ( 4 X - 2 )\) Circle your answer.
2426102104
AQA Further AS Paper 2 Statistics 2021 June Q2
1 marks Easy -1.2
2 The random variable \(A\) has a Poisson distribution with mean 2 The random variable \(B\) has a Poisson distribution with standard deviation 4 The random variables \(A\) and \(B\) are independent.
State the distribution of \(A + B\) Circle your answer.
[0pt] [1 mark]
Po(4)
Po(6)
Po(8)
Po(18)
AQA Further AS Paper 2 Statistics 2021 June Q3
5 marks Moderate -0.8
3 The random variable \(X\) has a discrete uniform distribution and takes values \(1,2,3 , \ldots , n\) The mean of \(X\) is 8 3
  1. Show that \(n = 15\) [0pt] [2 marks]
    LL
    3
  2. \(\quad\) Find \(\mathrm { P } ( X > 4 )\) 3
  3. Find the variance of \(X\), giving your answer in exact form.
AQA Further AS Paper 2 Statistics 2021 June Q4
7 marks Standard +0.3
4 The distance a particular football player runs in a match is modelled by a normal distribution with standard deviation 0.3 kilometres. A random sample of \(n\) matches is taken.
The distance the player runs in this sample of matches has mean 10.8 kilometres.
The sample is used to construct a \(93 \%\) confidence interval for the mean, of width 0.0543 kilometres, correct to four decimal places. 4
  1. Find the value of \(n\) 4
  2. Find the \(93 \%\) confidence interval for the mean, giving the limits to three decimal places.
    4
  3. Alison claims that the population mean distance the player runs is 10.7 kilometres. She carries out a hypothesis test at the 7\% level of significance using the random sample and the hypotheses $$\begin{aligned} & \mathrm { H } _ { 0 } : \mu = 10.7 \\ & \mathrm { H } _ { 1 } : \mu \neq 10.7 \end{aligned}$$ 4 (c) (i) State, with a reason, whether the null hypothesis will be accepted or rejected. 4 (c) (ii) Describe, in the context of the hypothesis test in part (c)(i), what is meant by a Type II error. \includegraphics[max width=\textwidth, alt={}, center]{9be40ed6-6df8-426a-8afd-fefc17287de6-06_2488_1730_219_141}
AQA Further AS Paper 2 Statistics 2021 June Q5
5 marks Easy -1.2
5 In a game it is known that:
  • 25\% of players score 0
  • 30\% of players score 5
  • 35\% of players score 10
  • 10\% of players score 20
Players receive prize money, in pounds, equal to 100 times their score.
5
  1. State the modal score.
    [0pt] [1 mark] 5
  2. Find the median score.
    5
  3. Find the mean prize money received by a player.
AQA Further AS Paper 2 Statistics 2021 June Q6
11 marks Standard +0.3
6 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 1 } { 114 } ( 4 x + 7 ) & 0 \leq x \leq 6 \\ 0 & \text { otherwise } \end{cases}$$ 6
  1. Show that the median of \(X\) is 3.87, correct to three significant figures.
    [0pt] [3 marks]
    6
  2. Find the exact value of \(\mathrm { P } ( X > 2 )\)
    6
  3. The continuous random variable \(Y\) has probability density function \(g ( y ) = \begin{cases} \frac { 1 } { 2 } y ^ { 2 } - \frac { 1 } { 6 } y ^ { 3 }1 \leq y \leq 3
    0\text { otherwise } \end{cases}\)
    "
    6 (c) (i) Show that \(\operatorname { Var } \left( \frac { 1 } { Y } \right) = \frac { 2 } { 81 }\)
    		\multirow[b]{2}{*}{
    [4 marks]
    [4 marks]
    }
AQA Further AS Paper 2 Statistics 2021 June Q7
11 marks Standard +0.3
7 Two employees, \(A\) and \(B\), both produce the same toy for a company. The company records the total number of errors made per day by each employee during a 40-day period. The results are summarised in the following table. Employee
Number of errors made per day
0123 or moreTotal
\(A\)81020240
B18415340
Total261435580
The company claims that there is an association between employee and number of errors made per day. 7
  1. Test the company's claim, using the \(5 \%\) level of significance.
    7
  2. By considering observed and expected frequencies, interpret in context the association between employee and number of errors made per day. \includegraphics[max width=\textwidth, alt={}, center]{9be40ed6-6df8-426a-8afd-fefc17287de6-12_2492_1723_217_150}
    \includegraphics[max width=\textwidth, alt={}]{9be40ed6-6df8-426a-8afd-fefc17287de6-16_2496_1721_214_148}
AQA Further AS Paper 2 Statistics Specimen Q1
1 marks Easy -1.2
1 The random variable \(T\) has probability density defined by $$\mathrm { f } ( t ) = \left\{ \begin{array} { c c } \frac { t } { 8 } & 0 \leq t \leq k \\ 0 & \text { otherwise } \end{array} \right.$$ Find the value of \(k\) [0pt] [1 mark] $$\begin{array} { l l l l } \frac { 1 } { 16 } & \frac { 1 } { 4 } & 4 & 16 \end{array}$$
AQA Further AS Paper 2 Statistics Specimen Q2
1 marks Easy -2.0
2 The discrete random variable \(X\) has probability distribution defined by $$\mathrm { P } ( X = x ) = \begin{cases} 0.1 & x = 0,1,2,3,4,5,6,7,8,9 \\ 0 & \text { otherwise } \end{cases}$$ Find the value of \(\mathrm { P } ( 4 \leq X \leq 7 )\) Circle your answer.
0.20.30.40.5
AQA Further AS Paper 2 Statistics Specimen Q3
4 marks Standard +0.3
3 The discrete random variable \(R\) has the following probability distribution.
\(\boldsymbol { r }\)- 20\(a\)4
\(\mathbf { P } ( \boldsymbol { R } = \boldsymbol { r } )\)0.3\(b\)\(c\)0.1
It is known that \(\mathrm { E } ( R ) = 0.2\) and \(\operatorname { Var } ( R ) = 3.56\) Find the values of \(a , b\) and \(c\).
[0pt] [4 marks]
AQA Further AS Paper 2 Statistics Specimen Q4
3 marks Moderate -0.3
4 The number of printers, \(V\), bought during one day from the Verigood store can be modelled by a Poisson distribution with mean 4.5 The number of printers, \(W\), bought during one day from the Winnerprint store can be modelled by a Poisson distribution with mean 5.5 4
  1. Find the probability that the total number of printers bought during one day from Verigood and Winnerprint stores is greater than 10.
    [0pt] [2 marks] 4
  2. State the circumstance under which the distributional model you used in part (a) would not be valid.
    [0pt] [1 mark]
AQA Further AS Paper 2 Statistics Specimen Q5
5 marks Standard +0.8
5 Participants in a school jumping competition gain a total score for each jump based on the length, \(L\) metres, jumped beyond a fixed point and a mark, \(S\), for style. \(L\) may be regarded as a continuous random variable with probability density function $$\mathrm { f } ( l ) = \left\{ \begin{array} { c c } w l & 0 \leq l \leq 15 \\ 0 & \text { otherwise } \end{array} \right.$$ where \(w\) is a constant. \(S\) may be regarded as a discrete random variable with probability function $$\mathrm { P } ( S = s ) = \left\{ \begin{array} { c l } \frac { 1 } { 15 } s & s = 1,2,3,4,5 \\ 0 & \text { otherwise } \end{array} \right.$$ Assume that \(L\) and \(S\) are independent. The total score for a participant in this competition, \(T\), is given by \(T = L ^ { 2 } + \frac { 1 } { 2 } S\) Show that the expected total score for a participant is \(114 \frac { 1 } { 3 }\)
AQA Further AS Paper 2 Statistics Specimen Q6
8 marks Moderate -0.3
6 The continuous random variable \(T\) has probability density function defined by $$\mathrm { f } ( t ) = \left\{ \begin{array} { c c } \frac { 1 } { 3 } & 0 \leq t \leq \frac { 3 } { 2 } \\ \frac { 9 - 2 t } { 18 } & \frac { 3 } { 2 } \leq t \leq \frac { 9 } { 2 } \\ 0 & \text { otherwise } \end{array} \right.$$ 6
    1. Sketch this probability density function below. \includegraphics[max width=\textwidth, alt={}, center]{6ccf7d1d-5a7b-47d1-b38e-c7e762204746-07_1009_1041_1073_520} 6
      1. (ii) State the median of \(T\). 6
      1. Find \(\mathrm { E } ( T )\) [0pt] [2 marks]
        6
    3. (ii) Given that \(\mathrm { E } \left( T ^ { 2 } \right) = \frac { 15 } { 4 }\), find \(\operatorname { Var } ( 4 T - 5 )\) [3 marks]
AQA Further AS Paper 2 Statistics Specimen Q7
9 marks Standard +0.3
7 A dairy industry researcher, Robyn, decided to investigate the milk yield, classified as low, medium or high, obtained from four different breeds of cow, \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D . The milk yield of a sample of 105 cows was monitored and the results are summarised in contingency Table 1.
\multirow{2}{*}{Table 1}Yield
LowMediumHighTotal
\multirow{4}{*}{Breed}A451221
B106420
C817732
D520732
Total274830105
The sample of cows may be regarded as random.
Robyn decides to carry out a \(\chi ^ { 2 }\)-test for association between milk yield and breed using the information given in Table 1. 7
  1. Contingency Table 2 gives some of the expected frequencies for this test.
    Complete Table 2 with the missing expected values.
    \multirow[t]{2}{*}{Table 2}Yield
    LowMediumHigh
    \multirow{4}{*}{Breed}A6
    B5.149.145.71
    C
    D8.2314.639.14
    7
  2. (i) For Robyn's test, the test statistic \(\sum \frac { ( O - E ) ^ { 2 } } { E } = 19.4\) correct to three significant figures.
    Use this information to carry out Robyn's test, using the \(1 \%\) level of significance.
    7 (b) (ii) By considering the observed frequencies given in Table 1 with the expected frequencies in Table 2, interpret, in context, the association, if any, between milk yield and breed.
AQA Further AS Paper 2 Statistics Specimen Q8
9 marks Standard +0.3
8 In a small town, the number of properties sold during a week in spring by a local estate agent, Keith, can be regarded as occurring independently and with constant mean \(\mu\). Data from several years have shown the value of \(\mu\) to be 3.5 . A new housing development was built on the outskirts of the town and the properties on this development were offered for sale by the builder of the development, not by the local estate agents. During the first four weeks in spring, when properties on the new development were offered for sale by the builder, Keith sold a total of 8 properties. Keith claims that the sale of new properties by the builder reduced his mean number of properties sold during a week in spring. 8
  1. Investigate Keith's claim, using the \(5 \%\) level of significance.
    [0pt] [6 marks]
    8
  2. For your test carried out in part (a) state, in context, the meaning of a Type II error.
    [0pt] [1 mark]
    8
  3. State one advantage and one disadvantage of using a 1\% significance level rather than a 5\% level of significance in a hypothesis test.
    [0pt] [2 marks]
AQA Further AS Paper 2 Mechanics 2020 June Q1
1 marks Moderate -0.8
1 In this question use \(g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) A particle of mass 2 kg is attached to one end of a light elastic string of natural length 0.5 metres and modulus of elasticity 100 N . The other end of the string is attached to the point \(O\). Find the extension of the elastic string when the particle hangs in equilibrium vertically below \(O\). Circle your answer.
0.01 m
0.1 m
0.2 m
0.4 m
AQA Further AS Paper 2 Mechanics 2020 June Q2
1 marks Moderate -0.8
2 An object moves under the action of a single force \(F\) newtons.
It is given that \(F = 6 x ^ { 2 }\), where \(x\) represents the displacement in metres from the initial position of the object. Find the work done by \(F\) in moving the object from \(x = 1\) to \(x = 2\) Circle your answer.
[0pt] [1 mark]
12 J
14 J
18J
42 J