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Trigonometric modelling

Past-paper questions pack

Name
Class
Date set
Date due
Score/ 33

Questions are numbered; the mark schemes (the answer key) follow on separate pages so the question pages can be handed out alone. www.mathsaleveltutor.com

Edexcel AS and A-level Maths past-paper questions since June 2018 · 14 parts, 33 marks · mark schemes follow the questions · www.mathsaleveltutor.com · Questions and mark schemes © Pearson Education Ltd

1. June 2024 · Paper 1 · Q12(b) · 2 marks
A scientist studies the number of rabbits and the number of foxes in a wood for one year. The number of rabbits, R, is modelled by the equation R = A + 140 cos(30t)° – 480 sin(30t)° where t months is the time after the start of the year and A is a constant. Given that, during the year, the maximum number of rabbits in the wood is 1500 (b) (i) find a complete equation for this model. (ii) Hence write down the minimum number of rabbits in the wood during the year according to the model. (2)
2. June 2024 · Paper 1 · Q12(c) · 2 marks
The actual number of rabbits in the wood is at its minimum value in the middle of April. (c) Use this information to comment on the model for the number of rabbits. (2)
3. June 2024 · Paper 1 · Q12(d) · 4 marks
The number of foxes, F, in the wood during the same year is modelled by the equation F = 100 + 70 sin(30t + 70)° The number of foxes is at its minimum value after T months. (d) Find, according to the models, the number of rabbits in the wood at time T months. (4)
4. June 2023 · Paper 1 · Q13(c) · 2 marks
In an alternative model, the vertical height, H m, of a carriage above the ground, t seconds after the carriage starts the first circuit, is given by H = 29 cos(9t + alpha )° + beta 0 alpha < 360° where alpha and beta are constants. (c) Find a complete equation for the alternative model. (2)
5. June 2023 · Paper 1 · Q13(d) · 1 marks
Given that the carriage moves continuously for 2 minutes, (d) give a reason why the alternative model would be more appropriate. (1)
6. June 2022 · Paper 2 · Q9(a) · 4 marks
9. H H t Figure 4 Figure 5 Figure 4 shows a sketch of a Ferris wheel. The height above the ground, H m, of a passenger on the Ferris wheel, t seconds after the wheel starts turning, is modelled by the equation H = ½A sin(bt + alpha )°½ where A, b and alpha are constants. Figure 5 shows a sketch of the graph of H against t, for one revolution of the wheel. Given that • the maximum height of the passenger above the ground is 50 m • the passenger is 1 m above the ground when the wheel starts turning • the wheel takes 720 seconds to complete one revolution (a) find a complete equation for the model, giving the exact value of A, the exact value of b and the value of alpha to 3 significant figures. (4)
7. June 2022 · Paper 2 · Q9(b) · 1 marks
(b) Explain why an equation of the form H = ½A sin(bt + alpha )°½ + d where d is a positive constant, would be a more appropriate model. (1)
8. Oct 2021 · Paper 2 · Q15(b) · 3 marks
P C H metres Water level Figure 6 Figure 6 shows the cross-section of a water wheel. The wheel is free to rotate about a fixed axis through the point C. The point P is at the end of one of the paddles of the wheel, as shown in Figure 6. The water level is assumed to be horizontal and of constant height. The vertical height, H metres, of P above the water level is modelled by the equation H = 3 + 4 cos (0.5t) – 2 sin (0.5t) where t is the time in seconds after the wheel starts rotating. Using the model, find (b) (i) the maximum height of P above the water level, (ii) the value of t when this maximum height first occurs, giving your answer to one decimal place. (3)
9. Oct 2021 · Paper 2 · Q15(c) · 4 marks
In a single revolution of the wheel, P is below the water level for a total of T seconds. According to the model, (c) find the value of T giving your answer to 3 significant figures. (Solutions based entirely on calculator technology are not acceptable.) (4)
10. Oct 2021 · Paper 2 · Q15(d) · 1 marks
In reality, the water level may not be of constant height. (d) Explain how the equation of the model should be refined to take this into account. (1)
11. Oct 2020 · Paper 1 · Q6(b) · 1 marks
The temperature, theta °C , inside a room on a given day is modelled by the equation pi t pi t - - 0 t < 24 theta = 5 + sin + 2 cos where t is the number of hours after midnight. Using the equation of the model and your answer to part (a), (b) deduce the maximum temperature of the room during this day, (1)
12. Oct 2020 · Paper 1 · Q6(c) · 3 marks
(c) find the time of day when the maximum temperature occurs, giving your answer to the nearest minute. (3)
13. June 2018 · Paper 1 · Q8(a) · 1 marks
8. The depth of water, D metres, in a harbour on a particular day is modelled by the formula D = 5 + 2 sin (30t)° 0 t < 24 where t is the number of hours after midnight. A boat enters the harbour at 6:30 am and it takes 2 hours to load its cargo. The boat requires the depth of water to be at least 3.8 metres before it can leave the harbour. (a) Find the depth of the water in the harbour when the boat enters the harbour. (1)
14. June 2018 · Paper 1 · Q8(b) · 4 marks
(b) Find, to the nearest minute, the earliest time the boat can leave the harbour. (Solutions based entirely on graphical or numerical methods are not acceptable.) (4)

Mark schemes

Each mark scheme shows the whole question; the parts on this topic are the ones listed above.

June 2024 · Paper 1 · Q12
Mark scheme June 2024 Paper 1 Q12(d)
June 2023 · Paper 1 · Q13
Mark scheme June 2023 Paper 1 Q13(d)
June 2022 · Paper 2 · Q9
Mark scheme June 2022 Paper 2 Q9(b)
Oct 2021 · Paper 2 · Q15
Mark scheme Oct 2021 Paper 2 Q15(d)
Oct 2020 · Paper 1 · Q6
Mark scheme Oct 2020 Paper 1 Q6(c)
June 2018 · Paper 1 · Q8
Mark scheme June 2018 Paper 1 Q8(b)