plumiFlows, game theory and dynamic programming: Edexcel A‑Level Further Maths questions
Every Edexcel A-Level Further Maths exam question on D.6 Flows, game theory and dynamic programming, 2018–2025, with its mark scheme and examiner’s report.
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All 51 questions on Flows, game theory and dynamic programming, as a list
- June 2025, Decision 2, Q1 (3 marks): Ravi can choose one of three options, A, B or C, when playing a game. The profit, in pounds, associated with each outcome, and the corresponding probabilities,
- June 2025, Decision 2, Q4 (11 marks): A E F C1 C1 C2 C2 D G C T S (20, 40) (2, 33) (16, 35) (6, 13) (5, 12) (0, 11) (3, 19) (2, 6) (31, 43) (3, 5) (13, 27) (2, 9) (3, 16) (2, 15) (2, 6) (20, 25) (0,
- June 2025, Decision 2, Q5 (13 marks): Anvi makes boats during the winter months. She can make up to five boats each month. If she builds more than three boats in any one month, she must hire an assi
- June 2025, Decision 2, Q6 (15 marks): Player B Option P Option Q Option R Player A Option X 4 –2 –5 Option Y –1 1 3 A two person zero-sum game is represented by the pay-off matrix for player A shown
- June 2025, Decision 2, Q7 (14 marks): A sequence {un }, where n 0, satisfies the first order recurrence relation (*) un + 1 = αun – β(k n) (*) where α, β, k are non-zero constants and α ≠ k (a) De
- June 2025, AS Decision 2, Q2 (11 marks): Figure 1 Figure 1 shows a capacitated, directed network of pipes. The number on each arc represents the capacity of the corresponding pipe. The numbers in circl
- June 2025, AS Decision 2, Q3 (11 marks): Layla and Mohsin play a zero-sum game represented by the following pay-off matrix for Layla. Mohsin plays X Layla plays P Layla plays Q Layla plays R Mohsin pla
- June 2025, AS Decision 2, Q4 (9 marks): A sequence{ } un , where n 1, satisfies the recurrence relation 4 2 3 5 1 u u n n n Given that u1 2 (a) solve the recurrence relation, giving un in terms of n
- June 2024, Decision 2, Q1 (10 marks): 2 2 13 x 12 12 15 10 19 7 15 8 13 13 5 5 4 4 17 11 16 16 10 4 11 8 12 y 7 5 C1 A D E F T S B C C1 Figure 1 Figure 1 shows a capacitated, directed network of pip
- June 2024, Decision 2, Q2 (3 marks): The general solution of the first order recurrence relation u au bn cn d n n 1 2 n 0 is given by u A n n n ( ) 3 5 1 2 where A is an arbitrary non‑zero consta
- June 2024, Decision 2, Q5 (10 marks): Sebastien needs to make a journey. He can choose between travelling by plane, by train or by coach. Sebastien knows the exact costs of all three travel options,
- June 2024, Decision 2, Q6 (10 marks): A 3 3 3 3 2 2 4 5 5 6 5 7 1 4 4 4 4 2 4 D E F G I H J T S B C 3 Figure 2 The staged, directed network in Figure 2 represents the roads that connect 12 towns, S,
- June 2024, Decision 2, Q7 (13 marks): Player B Option X Option Y Option Z Option R 3 2 –3 Player A Option S 4 –2 1 Option T –1 3 6 A two person zero‑sum game is represented by the pay‑off matrix for
- June 2024, Decision 2, Q8 (8 marks): A sequence {un}, where n 0, satisfies the recurrence relation 2 5 3 8 2 2 1 u u u n n n n (a) Find the general solution of this recurrence relation. (5) A par
- June 2024, AS Decision 2, Q1 (8 marks): S A C G K T J F B H D E C1 C1 C2 C2 35 35 39 15 16 16 19 18 19 12 24 0 24 18 6 6 17 26 10 36 6 48 11 37 21 19 40 10 12 42 27 26 48 24 26 21 18 18 17 14 24 19 19
- June 2024, AS Decision 2, Q3 (14 marks): Haruki and Meera play a zero‑sum game. The game is represented by the following pay‑off matrix for Haruki. Meera Option X Option Y Option Z Haruki Option A 4 –2
- June 2024, AS Decision 2, Q4 (10 marks): Peter sets up a savings plan. He makes an initial deposit of £D and then pays in £M at the end of each month. The value of the savings plan, in pounds, is model
- June 2023, Decision 2, Q1 (7 marks): C2 C2 C1 C1 A F T E C S B D 12 17 10 10 4 52 29 4 5 5 8 8 2 6 22 18 18 11 11 10 17 17 15 20 26 19 21 44 Figure 1 Figure 1 shows a capacitated, directed network
- June 2023, Decision 2, Q2 (5 marks): An outdoor theatre is holding a summer gala performance. The theatre owner must decide whether to take out insurance against rain for this performance. The thea
- June 2023, Decision 2, Q5 (8 marks): A sequence un ,where n 0, satisfies the second order recurrence relation u u u n n n 2 1 1 2 3 where u u 0 1 15 20 = = (a) By considering the sequence vn , whe
- June 2023, Decision 2, Q6 (13 marks): Polly is a motivational speaker who is planning her engagements for the next four weeks. Polly will • visit four different countries in these four weeks • visit
- June 2023, Decision 2, Q7 (8 marks): Martina decides to open a bank account to help her to save for a holiday. Each month she puts £k into the account and allows herself to spend one quarter of wha
- June 2023, Decision 2, Q8 (17 marks): A two-person zero-sum game is represented by the pay-off matrix for player A shown below. Player A Player B Option X Option Q Option R Option S Option T −3 2 4
- June 2023, AS Decision 2, Q2 (9 marks): S A B C D E F G H 7 15 32 18 8 18 18 17 1 39 5 6 4 2 3 21 0 2 0 0 7 28 36 64 0 4 0 9 0 31 0 0 8 8 5 5 J T 2 13 Figure 1 An engineer monitors a system of pipes t
- June 2023, AS Decision 2, Q3 (14 marks): A two‑person zero‑sum game is represented by the following pay‑off matrix for player A. B plays X B plays Y A plays Q 2 –2 A plays R –1 5 A plays S 3 4 A plays
- June 2023, AS Decision 2, Q4 (8 marks): A sequence {un}, where n 0, satisfies the recurrence relation un + 1 = 3 2 un – 2n 2 – 4 u0 = k where k is an integer. (a) Determine an expression for un in t
- June 2022, Decision 2, Q2 (4 marks): The general solution of the second order recurrence relation un + 2 + k1un + 1 + k2un = 0 n 0 is given by un = (A + Bn)(–3)n where A and B are arbitrary non‑z
- June 2022, Decision 2, Q3 (7 marks): The table below shows the transport options, usual travel times, possible delay times and corresponding probabilities of delay for a journey. All times are in m
- June 2022, Decision 2, Q4 (9 marks): C1 20 33 41 30 17 6 14 22 53 75 24 8 4 14 30 A B C D S T E F G C1 C2 C2 20 29 41 25 15 6 14 22 42 72 19 5 4 9 30 10 12 Figure 1 Figure 1 shows a capacitated, di
- June 2022, Decision 2, Q6 (14 marks): Bernie makes garden sheds. He can build up to four sheds each month. If he builds more than two sheds in any one month, he must hire an additional worker at a c
- June 2022, Decision 2, Q7 (17 marks): Player B Option W Option X Option Y Option Z Player A Option Q 4 3 –1 –2 Option R –3 5 –4 k Option S –1 6 3 –3 A two person zero‑sum game is represented by the
- June 2022, Decision 2, Q8 (9 marks): The owner of a new company models the number of customers that the company will have at the end of each month. The owner assumes that • a constant proportion, p
- June 2022, AS Decision 2, Q2 (9 marks): C C2 C2 C1 C1 S F H A D G T B E 18 32 13 23 8 6 5 6 6 12 19 13 16 3 42 10 10 37 5 18 27 9 23 5 6 4 6 2 10 16 8 3 37 10 10 31 5 10 Figure 1 Figure 1 shows a capa
- June 2022, AS Decision 2, Q3 (14 marks): Terry and June play a zero‑sum game. The pay‑off matrix shows the number of points that Terry scores for each combination of strategies. June Option X Option Y
- June 2022, AS Decision 2, Q4 (10 marks): A sequence {un}, where n 0, satisfies the recurrence relation un + 1 + 3un = n + k where k is a non‑zero constant. Given that u0 = 1 (a) solve the recurrence
- November 2020, Decision 2, Q2 (7 marks): Jenny can choose one of three options, A, B or C, when playing a game. The profit, in pounds, associated with each outcome and their corresponding probabilities
- November 2020, Decision 2, Q4 (8 marks): The complementary function for the second order recurrence relation un + 2 + αun + 1 + βun = 20 (−3) n n 0 is given by un = A (2) n + B (−1) n where A and B a
- November 2020, Decision 2, Q5 (10 marks): C1 C1 A D C B E F G H J (5, 8) (5, 7) (6, 10) (1, 2) (0, 2) (4, 6) (4, 8) (1, 6) (1, 1) (0, 4) (2, 3) (3, 6) (4, 6) (3, 7) (4, 5) (2, 5) (5, 7) Figure 2 Figure
- November 2020, Decision 2, Q6 (14 marks): Player B Option X Option Y Option Z Player A Option Q 1 5 3 Option R 4 −3 1 Option S 2 −4 −2 Option T 3 −2 0 A two person zero-sum game is represented by the pa
- November 2020, Decision 2, Q7 (12 marks): A manufacturer can export five batches of footwear each year. Each exported batch contains just one type of footwear. The types of footwear are trainers, sandal
- November 2020, AS Decision 2, Q1 (9 marks): S A D B C C1 C2 F E T 15 15 29 26 23 y 26 23 17 7 18 15 13 10 8 5 37 32 18 18 12 12 7 0 14 x 12 10 C1 C2 Figure 1 Figure 1 shows a capacitated, directed network
- November 2020, AS Decision 2, Q3 (14 marks): Two teams, A and B, each have three team members. One member of Team A will compete against one member of Team B for 10 rounds of a competition. None of the rou
- November 2020, AS Decision 2, Q4 (8 marks): A sequence {un}, where n 1, satisfies the recurrence relation 2un = un – 1 – kn2 where 4u2 – u0 = 27k2 and k is a non‑zero constant. Show that, as n becomes la
- June 2019, Decision 2, Q3 (13 marks): B 24 24 14 –24 –24 –30 –17 –25 29 20 19 27 32 17 D C E F 19 G A H S I T J 13 28 13 x 18 Figure 1 In Figure 1 the weight of arc SB is denoted by x where x . 0 (a
- June 2019, Decision 2, Q4 (14 marks): Player B Option X Option Y Option Z Option P 3 –2 0 Player A Option Q –4 4 –2 Option R 1 2 –1 A two person zero-sum game is represented by the pay-off matrix fo
- June 2019, Decision 2, Q5 (11 marks): An increasing sequence {un} for n ` is such that the difference between the nth term of {un} and the mean of the previous two terms of {un} is always 6 (a) Sh
- June 2019, Decision 2, Q6 (8 marks): B H C2 C2 C1 C1 D E F C G A S T (3, 6) (5, 7) (4, 7) (5, 8) (6, 8) (5, 6) (4, 7) (17, 20) (10, 12) (10, 14) (2, 4) (13, 17) (8, 11) (7, 9) (0, 4) (3, 5) (3, 5)
- June 2019, Decision 2, Q7 (12 marks): Aisha is deciding whether or not to play a game. The game involves rolling three fair six-sided dice, which have faces numbered from 1 to 6 If the total score o
- June 2019, AS Decision 2, Q2 (6 marks): (a) Find the general solution of the recurrence relation un + 1 = 3un + 2n n . 1 (4) (b) Find the particular solution of this recurrence relation for which u1 =
- June 2019, AS Decision 2, Q3 (10 marks): 14 A B C D E F G H J S T 18 4 6 7 24 2 7 2 3 0 11 5 3 0 5 19 14 7 26 0 2 6 5 6 12 0 12 2 6 0 4 5 17 21 4 0 7 Figure 1 Alexa is monitoring a system of pipes thro
- June 2019, AS Decision 2, Q4 (15 marks): The table below gives the pay-off matrix for a zero-sum game between two players, Aljaz and Brendan. The values in the table show the pay-offs for Aljaz. Brenda
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