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AQA Further AS Paper 2 Discrete Specimen Q1
1 marks Easy -1.8
1 A graph has 5 vertices and 6 edges.
Find the sum of the degrees of the vertices. Circle your answer. 10111215
AQA Further AS Paper 2 Discrete Specimen Q2
1 marks Moderate -0.5
2 A connected planar graph has \(x\) vertices and \(2 x - 4\) edges.
Find the number of faces of the planar graph in terms of \(x\).
Circle your answer. \(x - 6\) \(x - 2\) \(6 - x\) \(2 - x\)
AQA Further AS Paper 2 Discrete Specimen Q3
2 marks Standard +0.8
3 The function min \(( a , b )\) is defined by: $$\begin{aligned} \min ( a , b ) & = a , a < b \\ & = b , \text { otherwise } \end{aligned}$$ For example, \(\min ( 7,2 ) = 2\) and \(\min ( - 4,6 ) = - 4\). Gary claims that the binary operation \(\Delta\), which is defined as $$x \Delta y = \min ( x , y - 3 )$$ where \(x\) and \(y\) are real numbers, is associative as finding the smallest number is not affected by the order of operation. Disprove Gary's claim.
[0pt] [2 marks]
AQA Further AS Paper 2 Discrete Specimen Q4
6 marks Moderate -0.5
4 A communications company is conducting a feasibility study into the installation of underground television cables between 5 neighbouring districts. The length of the possible pathways for the television cables between each pair of districts, in miles, is shown in the table. The pathways all run alongside cycle tracks.
BillingeGarswoodHaydockOrrellUp Holland
Billinge-2.5***4.34.8
Garswood2.5-3.1***5.9
Haydock***3.1-6.77.8
Orrell4.3***6.7-2.1
Up Holland4.85.97.82.1-
4
  1. Give a possible reason, in context, why some of the table entries are labelled as ***. 4
  2. As part of the feasibility study, Sally, an engineer needs to assess each possible pathway between the districts. To do this, Sally decides to travel along every pathway using a bicycle, starting and finishing in the same district. From past experience, Sally knows that she can travel at an average speed of 12 miles per hour on a bicycle. Find the minimum time, in minutes, that it will take Sally to cycle along every pathway.
    [0pt] [4 marks]
AQA Further AS Paper 2 Discrete Specimen Q5
8 marks Moderate -0.8
5 Charlotte is visiting a city and plans to visit its five monuments: \(A , B , C , D\) and \(E\).
The network shows the time, in minutes, that a typical tourist would take to walk between the monuments on a busy weekday morning. \includegraphics[max width=\textwidth, alt={}, center]{ba9e9840-ce27-4ca7-ab05-50461d135445-06_902_1134_529_543} Charlotte intends to walk from one monument to another until she has visited them all, before returning to her starting place. 5
  1. Use the nearest neighbour algorithm, starting from \(A\), to find an upper bound for the minimum time for Charlotte's tour.
    5
  2. By deleting vertex \(B\), find a lower bound for the minimum time for Charlotte's tour.
    [0pt] [3 marks]
    5
  3. Charlotte wants to complete the tour in 52 minutes. Use your answers to parts (a) and (b) to comment on whether this could be possible.
    5
  4. Charlotte takes 58 minutes to complete the tour. Evaluate your answers to part (a) and part (b) given this information.
    5
  5. Explain how this model for a typical tourist's tour may not be applicable if the tourist walked between the monuments during the evening.
AQA Further AS Paper 2 Discrete Specimen Q6
11 marks Standard +0.3
6 Victoria and Albert play a zero-sum game. The game is represented by the following pay-off matrix for Victoria.
\multirow{2}{*}{}Albert
Strategy\(\boldsymbol { x }\)\(Y\)\(z\)
\multirow{3}{*}{Victoria}\(P\)3-11
\(Q\)-201
\(R\)4-1-1
6
  1. Find the play-safe strategies for each player.
    6
  2. State, with a reason, the strategy that Albert should never play.
    6
  3. (i) Determine an optimal mixed strategy for Victoria.
    [0pt] [5 marks]
    6 (c) (ii) Find the value of the game for Victoria.
    6 (c) (iii) State an assumption that must made in order that your answer for part (c)(ii) is the maximum expected pay-off that Victoria can achieve.
AQA Further AS Paper 2 Discrete Specimen Q7
4 marks Moderate -0.5
7 The network shows a system of pipes, where \(S\) is the source and \(T\) is the sink.
The capacity, in litres per second, of each pipe is shown on each arc.
The cut shown in the diagram can be represented as \(\{ S , P , R \} , \{ Q , T \}\). \includegraphics[max width=\textwidth, alt={}, center]{ba9e9840-ce27-4ca7-ab05-50461d135445-10_629_1168_616_557} 7
  1. Complete the table below to give the value of each of the 8 possible cuts.
    CutValue
    \{ S \}\(\{ P , Q , R , T \}\)31
    \(\{ S , P \}\)\(\{ Q , R , T \}\)32
    \(\{ S , Q \}\)\(\{ P , R , T \}\)
    \(\{ S , R \}\)\(\{ P , Q , T \}\)
    \(\{ S , P , Q \}\)\(\{ R , T \}\)30
    \(\{ S , P , R \}\)\(\{ Q , T \}\)37
    \(\{ S , Q , R \}\)\(\{ P , T \}\)35
    \(\{ S , P , Q , R \}\)\(\{ T \}\)30
    7
  2. State the value of the maximum flow through the network. Give a reason for your answer.
    [0pt] [1 mark] 7
  3. Indicate on Figure 1 a possible flow along each arc, corresponding to the maximum flow through the network.
    [0pt] [2 marks] \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{ba9e9840-ce27-4ca7-ab05-50461d135445-11_618_1150_1260_557}
    \end{figure}
AQA Further AS Paper 2 Discrete Specimen Q8
8 marks Moderate -0.3
8 A family business makes and sells two kinds of kitchen table.
Each pine table takes 6 hours to make and the cost of materials is \(\pounds 30\).
Each oak table takes 10 hours to make and the cost of materials is \(\pounds 70\).
Each month, the business has 360 hours available for making the tables and \(\pounds 2100\) available for the materials.
Each month, the business sells all of its tables to a wholesaler.
The wholesaler specifies that it requires at least 10 oak tables per month and at least as many pine tables as oak tables. Each pine table sold gives the business a profit of \(\pounds 40\) and each oak table sold gives the business a profit of \(\pounds 75\). Use a graphical method to find the number of each type of table the business should make each month, in order to maximise its total profit. Show clearly how you obtain your answer.
[0pt] [8 marks]
-T...,T-\includegraphics[max width=\textwidth, alt={}]{ba9e9840-ce27-4ca7-ab05-50461d135445-13_24_64_421_1323}-T.....T
T\multirow{2}{*}{}TolooloTTo
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-- →Tou------T ----,-\includegraphics[max width=\textwidth, alt={}]{ba9e9840-ce27-4ca7-ab05-50461d135445-13_27_77_934_1310}- -T\includegraphics[max width=\textwidth, alt={}]{ba9e9840-ce27-4ca7-ab05-50461d135445-13_47_169_898_1567}T
-\includegraphics[max width=\textwidth, alt={}]{ba9e9840-ce27-4ca7-ab05-50461d135445-13_24_147_950_738}-\includegraphics[max width=\textwidth, alt={}]{ba9e9840-ce27-4ca7-ab05-50461d135445-13_45_261_946_1475}T
--
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- - ---\multirow{4}{*}{}\includegraphics[max width=\textwidth, alt={}]{ba9e9840-ce27-4ca7-ab05-50461d135445-13_29_150_1450_1310}-\includegraphics[max width=\textwidth, alt={}]{ba9e9840-ce27-4ca7-ab05-50461d135445-13_24_169_1448_1567}T
-↓- -
- - - - - --T --
-\includegraphics[max width=\textwidth, alt={}]{ba9e9840-ce27-4ca7-ab05-50461d135445-13_44_183_1637_536}- --
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- 1
- 1
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-- - - -\includegraphics[max width=\textwidth, alt={}]{ba9e9840-ce27-4ca7-ab05-50461d135445-13_38_158_2143_729}---
--\includegraphics[max width=\textwidth, alt={}]{ba9e9840-ce27-4ca7-ab05-50461d135445-13_45_183_2281_536}------------ --T
AQA Further Paper 1 2020 June Q1
1 marks Easy -1.8
1 Which of the integrals below is not an improper integral?
Circle your answer. \(\int _ { 0 } ^ { \infty } e ^ { - x } d x\) \(\int _ { 0 } ^ { 2 } \frac { 1 } { 1 - x ^ { 2 } } \mathrm {~d} x\) \(\int _ { 0 } ^ { 1 } \sqrt { x } \mathrm {~d} x\) \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { x } } \mathrm {~d} x\)
AQA Further Paper 1 2020 June Q2
1 marks Easy -1.2
2 Which one of the matrices below represents a rotation of \(90 ^ { \circ }\) about the \(x\)-axis? Circle your answer.
[0pt] [1 mark] \(\left[ \begin{array} { c c c } 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & - 1 \end{array} \right]\) \(\left[ \begin{array} { c c c } - 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right]\) \(\left[ \begin{array} { l l l } 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array} \right]\) \(\left[ \begin{array} { c c c } 1 & 0 & 0 \\ 0 & 0 & - 1 \\ 0 & 1 & 0 \end{array} \right]\)
AQA Further Paper 1 2020 June Q3
1 marks Standard +0.3
3 The quadratic equation \(a x ^ { 2 } + b x + c = 0 ( a , b , c \in \mathbb { R } )\) has real roots \(\alpha\) and \(\beta\). One of the four statements below is incorrect. Which statement is incorrect? Tick ( \(\checkmark\) ) one box. \(c = 0 \Rightarrow \alpha = 0\) or \(\beta = 0\) □ \(c = a \Rightarrow \alpha\) is the reciprocal of \(\beta\) □ \(b < 0\) and \(c < 0 \Rightarrow \alpha > 0\) and \(\beta > 0\) □ \(b = 0 \Rightarrow \alpha = - \beta\) □
AQA Further Paper 1 2020 June Q4
6 marks Standard +0.8
4
  1. Express \(z ^ { 4 } - 2 z ^ { 3 } + p z ^ { 2 } + r z + 80\) as the product of two quadratic factors with real coefficients.
    [4 marks]
    4 It is given that \(1 - 3 \mathrm { i }\) is one root of the quartic equation
    堛的 增
    4
  2. Find the value of \(p\) and the value of \(r\).
AQA Further Paper 1 2020 June Q5
9 marks Standard +0.8
5
  1. Show that the equation of \(H _ { 1 }\) can be written in the form $$( x - 1 ) ^ { 2 } - \frac { y ^ { 2 } } { q } = r$$ where \(q\) and \(r\) are integers.
    5
  2. \(\quad \mathrm { H } _ { 2 }\) is the hyperbola $$x ^ { 2 } - y ^ { 2 } = 4$$ Describe fully a sequence of two transformations which maps the graph of \(H _ { 2 }\) onto the graph of \(H _ { 1 }\) [0pt] [4 marks]
AQA Further Paper 1 2020 June Q6
9 marks Standard +0.8
6 Let \(w\) be the root of the equation \(z ^ { 7 } = 1\) that has the smallest argument \(\alpha\) in the interval \(0 < \alpha < \pi\) 6
  1. Prove that \(w ^ { n }\) is also a root of the equation \(z ^ { 7 } = 1\) for any integer \(n\). 6
  2. Prove that \(1 + w + w ^ { 2 } + w ^ { 3 } + w ^ { 4 } + w ^ { 5 } + w ^ { 6 } = 0\) 6
  3. Show the positions of \(w , w ^ { 2 } , w ^ { 3 } , w ^ { 4 } , w ^ { 5 }\), and \(w ^ { 6 }\) on the Argand diagram below.
    [0pt] [2 marks] \includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-08_835_898_1802_571} 6
  4. Prove that $$\cos \frac { 2 \pi } { 7 } + \cos \frac { 4 \pi } { 7 } + \cos \frac { 6 \pi } { 7 } = - \frac { 1 } { 2 }$$
AQA Further Paper 1 2020 June Q7
7 marks Challenging +1.2
7 Three planes have equations $$\begin{aligned} ( 4 k + 1 ) x - 3 y + ( k - 5 ) z & = 3 \\ ( k - 1 ) x + ( 3 - k ) y + 2 z & = 1 \\ 7 x - 3 y + 4 z & = 2 \end{aligned}$$ 7
  1. The planes do not meet at a unique point.
    Show that \(k = 4.5\) is one possible value of \(k\), and find the other possible value of \(k\).
    7
  2. For each value of \(k\) found in part (a), identify the configuration of the given planes.
    In each case fully justify your answer, stating whether or not the equations of the planes form a consistent system.
    [4 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
AQA Further Paper 1 2020 June Q8
6 marks Standard +0.8
8 The three roots of the equation $$4 x ^ { 3 } - 12 x ^ { 2 } - 13 x + k = 0$$ where \(k\) is a constant, form an arithmetic sequence. Find the roots of the equation.
AQA Further Paper 1 2020 June Q9
13 marks Challenging +1.2
9 The function f is defined by $$f ( x ) = \frac { x ( x + 3 ) } { x + 4 } \quad ( x \in \mathbb { R } , x \neq - 4 )$$ 9
  1. Find the interval ( \(a , b\) ) in which \(\mathrm { f } ( x )\) does not take any values.
    Fully justify your answer.
    9
  2. Find the coordinates of the two stationary points of the graph of \(y = \mathrm { f } ( x )\) 9
  3. Show that the graph of \(y = \mathrm { f } ( x )\) has an oblique asymptote and find its equation.
    \section*{Question 9 continues on the next page} 9
  4. Sketch the graph of \(y = \mathrm { f } ( x )\) on the axes below.
    [0pt] [4 marks] \includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-16_1100_1100_406_470} \includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-17_2493_1732_214_139}
    1. Fird \(\begin{aligned} & \text { Do not write } \\ & \text { outside the } \end{aligned}\)
AQA Further Paper 1 2020 June Q10
10 marks Challenging +1.2
10
  1. Find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 2 y } { x } = \frac { x + 3 } { x ( x - 1 ) \left( x ^ { 2 } + 3 \right) } \quad ( x > 1 )$$ 10
  2. Find the particular solution for which \(y = 0\) when \(x = 3\) Give your answer in the form \(y = \mathrm { f } ( x )\)
AQA Further Paper 1 2020 June Q11
11 marks Standard +0.8
11 The lines \(l _ { 1 } , l _ { 2 }\) and \(l _ { 3 }\) are defined as follows. $$\begin{aligned} & l _ { 1 } : \left( \mathbf { r } - \left[ \begin{array} { c } 1 \\ 5 \\ - 1 \end{array} \right] \right) \times \left[ \begin{array} { c } - 2 \\ 1 \\ - 3 \end{array} \right] = \mathbf { 0 } \\ & l _ { 2 } : \left( \mathbf { r } - \left[ \begin{array} { c } - 3 \\ 2 \\ 7 \end{array} \right] \right) \times \left[ \begin{array} { c } 2 \\ - 1 \\ 3 \end{array} \right] = \mathbf { 0 } \\ & l _ { 3 } : \left( \mathbf { r } - \left[ \begin{array} { c } - 5 \\ 12 \\ - 4 \end{array} \right] \right) \times \left[ \begin{array} { l } 4 \\ 0 \\ 9 \end{array} \right] = \mathbf { 0 } \end{aligned}$$ 11
    1. Explain how you know that two of the lines are parallel.
      11
      1. (ii)
      Show that the perpendicular distance between these two parallel lines is 7.95 units, correct to three significant figures.
      [5 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
      11
  2. Show that the lines \(l _ { 1 }\) and \(l _ { 3 }\) meet, and find the coordinates of their point of intersection. \includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-23_2488_1716_219_153}
AQA Further Paper 1 2020 June Q12
8 marks Standard +0.8
12
  1. Use the definition of the cosh function to prove that $$\cosh ^ { - 1 } \left( \frac { x } { a } \right) = \ln \left( \frac { x + \sqrt { x ^ { 2 } - a ^ { 2 } } } { a } \right) \quad \text { for } a > 0$$ [6 marks]
    12
  2. The formulae booklet gives the integral of \(\frac { 1 } { \sqrt { x ^ { 2 } - a ^ { 2 } } }\) as $$\cosh ^ { - 1 } \left( \frac { x } { a } \right) \text { or } \ln \left( x + \sqrt { x ^ { 2 } - a ^ { 2 } } \right) + c$$ Ronald says that this contradicts the result given in part (a).
    Explain why Ronald is wrong.
AQA Further Paper 1 2020 June Q13
12 marks Standard +0.8
13 Two light elastic strings each have one end attached to a particle \(B\) of mass \(3 c \mathrm {~kg}\), which rests on a smooth horizontal table. The other ends of the strings are attached to the fixed points \(A\) and \(C\), which are 8 metres apart. \(A B C\) is a horizontal line. \includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-26_92_910_635_566} String \(A B\) has a natural length of 4 metres and a stiffness of \(5 c\) newtons per metre.
String \(B C\) has a natural length of 1 metre and a stiffness of \(c\) newtons per metre.
The particle is pulled a distance of \(\frac { 1 } { 3 }\) metre from its equilibrium position towards \(A\), and released from rest. 13
  1. Show that the particle moves with simple harmonic motion.
    13
  2. Find the speed of the particle when it is at a point \(P\), a distance \(\frac { 1 } { 4 }\) metre from the equilibrium position. Give your answer to two significant figures.
    [0pt] [4 marks]
AQA Further Paper 1 2020 June Q14
6 marks Challenging +1.2
14
  1. Given that $$\sinh ( A + B ) = \sinh A \cosh B + \cosh A \sinh B$$ express \(\sinh ( m + 1 ) x\) and \(\sinh ( m - 1 ) x\) in terms of \(\sinh m x , \cosh m x , \sinh x\) and \(\cosh x\) 14
  2. Hence find the sum of the series $$C _ { n } = \cosh x + \cosh 2 x + \cdots + \cosh n x$$ in terms of \(\sinh x , \sinh n x\) and \(\sinh ( n + 1 ) x\) Do not write \includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-30_2491_1736_219_139}
AQA Further Paper 2 2021 June Q1
1 marks Easy -1.8
1 Which of the following matrices is singular?
Circle your answer. \(\left[ \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right]\) \(\left[ \begin{array} { l l } 1 & 1 \\ 2 & 2 \end{array} \right]\) \(\left[ \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right]\) \(\left[ \begin{array} { c c } 1 & - 2 \\ 1 & 2 \end{array} \right]\)
AQA Further Paper 2 2021 June Q2
1 marks Easy -1.2
2 Find arg ( \(- 4 - 7 \mathrm { i }\) ) to the nearest degree.
Circle your answer.
[0pt] [1 mark] \(- 120 ^ { \circ }\) \(- 60 ^ { \circ }\) \(30 ^ { \circ }\) \(60 ^ { \circ }\)
AQA Further Paper 2 2021 June Q3
1 marks Easy -1.2
3 The line \(L\) has equation \(\mathbf { r } = \left[ \begin{array} { l } 3 \\ 2 \\ 0 \end{array} \right] + \lambda \left[ \begin{array} { c } - 1 \\ - 2 \\ 5 \end{array} \right]\) Which of the following lines is perpendicular to the line \(L\) ?
Tick \(( \checkmark )\) one box. $$\begin{aligned} & \mathbf { r } = \left[ \begin{array} { c } 2 \\ - 3 \\ 4 \end{array} \right] + \mu \left[ \begin{array} { c } 1 \\ 2 \\ - 5 \end{array} \right] \\ & \mathbf { r } = \left[ \begin{array} { l } 1 \\ 0 \\ 1 \end{array} \right] + \mu \left[ \begin{array} { c } 2 \\ - 3 \\ 1 \end{array} \right] \\ & \mathbf { r } = \left[ \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right] + \mu \left[ \begin{array} { l } 1 \\ 1 \\ 2 \end{array} \right] \\ & \mathbf { r } = \left[ \begin{array} { l } 0 \\ 3 \\ 2 \end{array} \right] + \mu \left[ \begin{array} { l } 4 \\ 3 \\ 2 \end{array} \right] \end{aligned}$$ □
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