: Edexcel past-paper questions pack
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Addition and double-angle formulae, harmonic form

Past-paper questions pack

Name
Class
Date set
Date due
Score/ 117

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 · 32 parts, 117 marks · mark schemes follow the questions · www.mathsaleveltutor.com · Questions and mark schemes © Pearson Education Ltd

1. June 2025 · Paper 1 · Q14(a) · 3 marks
14. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. (a) Show that sin(x + 30°) + 3 cos(x + 30°) ≡ 2 cos x (3)
2. June 2025 · Paper 1 · Q14(b) · 4 marks
(b) Hence solve, for 0 theta < 180° sin( theta + 30°) + 3 cos( theta + 30°) = 3 sin 2 theta giving your answers to one decimal place, where appropriate. (4)
3. June 2025 · Paper 2 · Q15(b) · 5 marks
(b) Using the substitution x = tan theta show that beta . / x x theta theta f d c os 2 d alpha where alpha and beta are constants to be found. (5)
4. June 2024 · Paper 1 · Q12(a) · 3 marks
12. (a) Express 140 cos theta – 480 sin theta in the form K cos( theta + alpha ) where K > 0 and 0 < alpha < 90° State the value of K and give the value of alpha , in degrees, to 2 decimal places. (3)
5. 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)
6. June 2024 · Paper 1 · Q13(a) · 4 marks
13. (a) Given that a is a positive constant, use the substitution x = a sin 2 theta to show that pi a : : x a x x a theta theta d sin 2 d (4)
7. June 2024 · Paper 1 · Q13(b) · 4 marks
(b) Hence use algebraic integration to show that a : x a x x k pi a d where k is a constant to be found. (4)
8. June 2023 · Paper 1 · Q12 · 5 marks
12. y = sin x where x is measured in radians. Use differentiation from first principles to show that d y x = cos x d You may • use without proof the formula for sin(A ± B) assume that as h → 0, sin h → 1 and cosh -1 → 0 • h h (5)
9. June 2023 · Paper 2 · Q8(a) · 3 marks
8. (a) Express 2 cos theta + 8 sin theta in the form R cos( theta – alpha ) , where R and alpha are constants, pi R > 0 and 0 < alpha < 2 Give the exact value of R and give the value of alpha in radians to 3 decimal places. (3)
10. June 2023 · Paper 2 · Q8(b) · 3 marks
The first three terms of an arithmetic sequence are cos x cos x + sin x cos x + 2 sin x x ≠ n pi Given that S9 represents the sum of the first 9 terms of this sequence as x varies, (b) (i) find the exact maximum value of S9 (ii) deduce the smallest positive value of x at which this maximum value of S9 occurs. (3)
11. June 2023 · Paper 2 · Q14(a) · 3 marks
14. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. (a) Show that the equation 2 tan theta (8 cos theta + 23 sin 2 theta ) = 8 sin 2 theta (1 + tan 2 theta ) may be written as 2 theta + B cos theta + C) = 0 sin 2 theta (A cos where A, B and C are constants to be found. (3)
12. June 2023 · Paper 2 · Q15 · 3 marks
15. A student attempts to answer the following question: Given that x is an obtuse angle, use algebra to prove by contradiction that sin x – cos x 1 The student starts the proof with: Assume that sin x – cos x < 1 when x is an obtuse angle Þ(sin x – cos x) 2 < 1 Þ... The start of the student’s proof is reprinted below. Complete the proof. (3)
13. June 2022 · Paper 1 · Q14(a) · 4 marks
14. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. (a) Given that 2 sin (x - 60°) = cos (x - 30°) show that tan x = 3 3 (4)
14. June 2022 · Paper 1 · Q14(b) · 4 marks
(b) Hence or otherwise solve, for 0 theta < 180° 2 sin 2 theta = cos (2 theta + 30°) giving your answers to one decimal place. (4)
15. June 2022 · Paper 1 · Q16(a) · 5 marks
16. y C R x Figure 6 Figure 6 shows a sketch of the curve C with parametric equations x = 8 sin2 t y = 2 sin 2t + 3 sin t 0 t pi The region R, shown shaded in Figure 6, is bounded by C, the x-axis and the line with equation x = 4 (a) Show that the area of R is given by a ∫ ( ) - cos t + sin t cos t d t where a is a constant to be found. (5)
16. Oct 2021 · Paper 1 · Q10(a) · 4 marks
10. In this question you should show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. (a) Given that 1 + cos 2 theta + sin 2 theta ≠ 0 prove that - theta + theta cos2 sin 2 ≡ theta tan theta theta + + cos2 sin 2 (4)
17. Oct 2021 · Paper 2 · Q15(a) · 3 marks
15. (a) Express 2cos theta – sin theta in the form R cos ( theta + alpha ), where R > 0 and 0 < alpha < pi Give the exact value of R and the value of alpha in radians to 3 decimal places. (3)
18. 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)
19. 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)
20. Oct 2020 · Paper 1 · Q6(a) · 3 marks
6. (a) Express sin x + 2 cos x in the form R sin (x + alpha ) where R and alpha are constants, R > 0 pi and 0 < alpha < 2 Give the exact value of R and give the value of alpha in radians to 3 decimal places. (3)
21. 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)
22. Oct 2020 · Paper 2 · Q10(a) · 4 marks
10. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. (a) Show that cos 3 A ≡ 4 cos3 A - 3 cos A (4)
23. Oct 2020 · Paper 2 · Q12(a)(i) · 3 marks
12. y R x Figure 3 The curve shown in Figure 3 has parametric equations pi x = 6 sin t y = 5 sin 2t 0 t 2 The region R, shown shaded in Figure 3, is bounded by the curve and the x-axis. pi ∫ (a) (i) Show that the area of R is given by 60sin cos t t t d (3)
24. June 2019 · Paper 1 · Q6(a) · 6 marks
6. (a) Solve, for –180° - ș - 180°, the equation 5 sin 2ș = 9 tan ș giving your answers, where necessary, to one decimal place. [Solutions based entirely on graphical or numerical methods are not acceptable.] (6)
25. June 2019 · Paper 2 · Q12(a) · 4 marks
12. (a) Prove cos3 theta + sin3 theta theta theta { 2௘cot 2ș ș z (90n)°, n  ] sin cos (4)
26. June 2019 · Paper 2 · Q12(b) · 3 marks
(b) Hence solve, for 90° : ș : 180°, the equation cos3 theta + sin3 theta theta theta = 4 sin cos giving any solutions to one decimal place. (3)
27. June 2018 · Paper 1 · Q5 · 5 marks
5. Given that theta pi pi 3sin y = - theta < < theta + theta 2sin 2cos show that y A pi pi d = - theta < < theta + theta d sin 2 where A is a rational constant to be found. (5)
28. June 2018 · Paper 1 · Q14(a) · 2 marks
14. A curve C has parametric equations x = 3 + 2 sin t, y = 4 + 2 cos 2t, 0 t < 2 pi (a) Show that all points on C satisfy y = 6 - (x - 3)2 (2)
29. June 2018 · Paper 2 · Q7(i) · 4 marks
pi , the equation 7. (i) Solve, for 0 x < 2 4 sin x = sec x (4)
30. June 2018 · Paper 2 · Q7(ii) · 5 marks
(ii) Solve, for 0 theta < 360°, the equation 5 sin theta - 5 cos theta = 2 giving your answers to one decimal place. (Solutions based entirely on graphical or numerical methods are not acceptable.) (5)
31. June 2018 · Paper 2 · Q9 · 5 marks
9. Given that theta is measured in radians, prove, from first principles, that d ( ) cos theta = - sin theta d theta You may assume the formula for cos (A ± B) and that as h → 0, sin h → 1 and cosh -1 → 0 h h (5)
32. June 2018 · Paper 2 · Q12(a) · 3 marks
12. (a) Prove that n pi 1 - cos 2 theta ≡ tan theta sin 2 theta , theta ≠ (2 + 1) , n ∈ (3)

Mark schemes

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

June 2025 · Paper 1 · Q14
Mark scheme June 2025 Paper 1 Q14(b)
June 2025 · Paper 2 · Q15
Mark scheme June 2025 Paper 2 Q15(b)
June 2024 · Paper 1 · Q12
Mark scheme June 2024 Paper 1 Q12(b)
June 2024 · Paper 1 · Q13
Mark scheme June 2024 Paper 1 Q13(b)
June 2023 · Paper 1 · Q12
Mark scheme June 2023 Paper 1 Q12
June 2023 · Paper 2 · Q8
Mark scheme June 2023 Paper 2 Q8(b)
June 2023 · Paper 2 · Q14
Mark scheme June 2023 Paper 2 Q14(a)
June 2023 · Paper 2 · Q15
Mark scheme June 2023 Paper 2 Q15
June 2022 · Paper 1 · Q14
Mark scheme June 2022 Paper 1 Q14(b)
June 2022 · Paper 1 · Q16
Mark scheme June 2022 Paper 1 Q16(a)
Oct 2021 · Paper 1 · Q10
Mark scheme Oct 2021 Paper 1 Q10(a)
Oct 2021 · Paper 2 · Q15
Mark scheme Oct 2021 Paper 2 Q15(c)
Oct 2020 · Paper 1 · Q6
Mark scheme Oct 2020 Paper 1 Q6(b)
Oct 2020 · Paper 2 · Q10
Mark scheme Oct 2020 Paper 2 Q10(a)
Oct 2020 · Paper 2 · Q12
Mark scheme Oct 2020 Paper 2 Q12(a)(i)
June 2019 · Paper 1 · Q6
Mark scheme June 2019 Paper 1 Q6(a)
June 2019 · Paper 2 · Q12
Mark scheme June 2019 Paper 2 Q12(b)
June 2018 · Paper 1 · Q5
Mark scheme June 2018 Paper 1 Q5
June 2018 · Paper 1 · Q14
Mark scheme June 2018 Paper 1 Q14(a)
June 2018 · Paper 2 · Q7
Mark scheme June 2018 Paper 2 Q7(ii)
June 2018 · Paper 2 · Q9
Mark scheme June 2018 Paper 2 Q9
June 2018 · Paper 2 · Q12
Mark scheme June 2018 Paper 2 Q12(a)