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Chain, product and quotient rules; connected rates of change

Past-paper questions pack

Name
Class
Date set
Date due
Score/ 138

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 · 36 parts, 138 marks · mark schemes follow the questions · www.mathsaleveltutor.com · Questions and mark schemes © Pearson Education Ltd

1. June 2025 · Paper 1 · Q9(b) · 4 marks
(b) show that the maximum speed of the car occurs when ; t = 5 ln :9 t 1 (4)
2. June 2025 · Paper 2 · Q5(c) · 3 marks
(c) Hence, or otherwise, find the gradient of C at the point P. (3)
3. June 2025 · Paper 2 · Q15(a) · 5 marks
15. y R x –1 P Q Figure 5 In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Figure 5 shows a sketch of part of the curve with equation y = f (x), where x . / O f x / x The curve • intersects the x‑axis at –1 and 1 • has minimum turning points at P and Q as shown in Figure 5. (a) Use calculus to find the exact coordinates of P. (5)
4. June 2024 · Paper 1 · Q5(a) · 3 marks
5. The function f is defined by x x < f(x) = x (a) Show that f′(x) = ax bx c / x where a, b and c are constants to be found. (3)
5. June 2024 · Paper 2 · Q6(a) · 2 marks
6. y y = f(x) y = g(x) x Figure 1 Figure 1 shows a sketch of the curves with equations y = f(x) and y = g(x) where 4x2 – 1 x > 0 f(x) = e g(x) = 8 ln x x > 0 (a) Find (i) f′(x) (ii) g′(x) (2)
6. June 2024 · Paper 2 · Q6(b) · 2 marks
Given that f′(x) = g′(x) at x = alpha (b) show that alpha satisfies the equation x x ln (2)
7. June 2024 · Paper 2 · Q10(a) · 5 marks
10. y C x Figure 4 Figure 4 shows a sketch of the curve C with parametric equations / x t y O 3 t t The point P with coordinates (4, 2) lies on C. (a) Using parametric differentiation, show that the tangent to C at P has equation x y (5)
8. June 2024 · Paper 2 · Q15(a) · 5 marks
15. The curve C has equation x y x y ( ) (a) Find an expression for d y x in terms of x and y. d (5)
9. June 2023 · Paper 1 · Q15(a) · 5 marks
15. A curve has equation y = f(x), where x x e f(x) = x > ln x - e (a) Show that x x x Ax B f′(x) = 7 e ( e ( ) ) x 2(e ) where A and B are constants to be found. (5)
10. June 2023 · Paper 1 · Q15(b) · 2 marks
(b) Hence show that the x coordinates of the turning points of the curve are solutions of the equation x e x x e (2)
11. June 2023 · Paper 2 · Q7(a) · 4 marks
7. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. A curve has equation x 3 + 2xy + 3y 2 = 47 (a) Find d y x in terms of x and y d (4)
12. June 2023 · Paper 2 · Q11(a) · 3 marks
11. h m 5 m 10 m 20 m Figure 1 A tank in the shape of a cuboid is being filled with water. The base of the tank measures 20 m by 10 m and the height of the tank is 5 m, as shown in Figure 1. At time t minutes after water started flowing into the tank the height of the water was h m and the volume of water in the tank was V m In a model of this situation • the sides of the tank have negligible thickness • the rate of change of V is inversely proportional to the square root of h (a) Show that h λ d = t d h where λ is a constant. (3)
13. June 2022 · Paper 1 · Q8(b) · 4 marks
(b) show that the maximum speed of the car occurs when t = - 1 + ln( t + ) (4)
14. June 2022 · Paper 2 · Q12(a) · 3 marks
12. The function f is defined by x e3 f(x) = x + k where k is a positive constant. (a) Show that fʹ(x) = (12x 2 - 8x + 3k) g(x) where g(x) is a function to be found. (3)
15. June 2022 · Paper 2 · Q16(a) · 5 marks
16. y C P l x Figure 6 Figure 6 shows a sketch of the curve C with parametric equations pi t 3 pi x = 2 tan t + 1 y = 2 sec 2 t + 3 - 4 pi The line l is the normal to C at the point P where t = 4 (a) Using parametric differentiation, show that an equation for l is y = - 1 2 x + 17 (5)
16. Oct 2021 · Paper 1 · Q4(a) · 4 marks
4. The curve with equation y = f (x) where f (x) = x2 + ln (2x2 - 4x + 5) has a single turning point at x = alpha (a) Show that alpha is a solution of the equation 2x3 - 4x2 + 7x - 2 = 0 (4)
17. Oct 2021 · Paper 1 · Q14 · 4 marks
14. Given that y = x - x > 0 + x show that d y x = A x x > 0 d where A is a constant to be found. (4)
18. Oct 2021 · Paper 2 · Q8(a) · 4 marks
8. The curve C has equation px3 + qxy + 3y2 = 26 where p and q are constants. (a) Show that y apx bqy d x qx cy d where a, b and c are integers to be found. (4)
19. Oct 2021 · Paper 2 · Q13(a) · 3 marks
13. The curve C has parametric equations x = sin 2 theta y = cosec3 theta 0 < theta < pi (a) Find an expression for d y x in terms of theta d (3)
20. Oct 2021 · Paper 2 · Q14(a) · 4 marks
14. 5 m h m T 3 m 8 m Figure 5 Water flows at a constant rate into a large tank. The tank is a cuboid, with all sides of negligible thickness. The base of the tank measures 8 m by 3 m and the height of the tank is 5 m. There is a tap at a point T at the bottom of the tank, as shown in Figure 5. At time t minutes after the tap has been opened • the depth of water in the tank is h metres • water is flowing into the tank at a constant rate of 0.48 m3 per minute • water is modelled as leaving the tank through the tap at a rate of 0.1h m3 per minute (a) Show that, according to the model, 1200 d h t = 24 – 5h d (4)
21. Oct 2020 · Paper 1 · Q9(a) · 3 marks
9. y C x Figure 2 Figure 2 shows a sketch of the curve C with equation y = f (x) where f (x) = 4(x2 - 2)e-2x x (a) Show that f ʹ(x) = 8(2 + x - x2)e-2x (3)
22. Oct 2020 · Paper 1 · Q14(a) · 3 marks
14. A large spherical balloon is deflating. At time t seconds the balloon has radius r cm and volume V cm3 The volume of the balloon is modelled as decreasing at a constant rate. (a) Using this model, show that d r t = -k d r 2 where k is a positive constant. (3)
23. Oct 2020 · Paper 1 · Q15(a) · 4 marks
15. The curve C has equation pi x2 tan y = 9 0 < y < 2 (a) Show that d y x = - x d x + (4)
24. Oct 2020 · Paper 1 · Q15(b) · 3 marks
(b) Prove that C has a point of inflection at x = (3)
25. Oct 2020 · Paper 2 · Q7(a) · 4 marks
7. y P x Figure 1 Figure 1 shows a sketch of the curve C with equation 2x x + y = 4 - 4 ln x x > 0 x (a) Show that d y 2x x x + - x = 12 d x x (4)
26. Oct 2020 · Paper 2 · Q13(b) · 3 marks
(b) Prove that gʹ (x) > 0 for all values of x in the domain of g. (3)
27. June 2019 · Paper 1 · Q3(a) · 4 marks
x + x 3. y = x ≠- x + ( ) y A (a) Show that d where A and n are constants to be found. = n x x + d ( )1 (4)
28. June 2019 · Paper 1 · Q12(a) · 4 marks
12. f(x) = 10e±௘0.25x sin x, x . 0 (a) Show that the x coordinates of the turning points of the curve with equation y = f(x) satisfy the equation tan x = 4 (4)
29. June 2019 · Paper 1 · Q14(a) · 2 marks
14. The curve C, in the standard Cartesian plane, is defined by the equation - pi pi x = 4 sin 2y = = = 4 < y < The curve C passes through the origin O (a) Find the value of d y x at the origin. d (2)
30. June 2019 · Paper 1 · Q14(c) · 3 marks
(c) Show that, for all points (x, y) lying on C, y d = x d a b - x where a and b are constants to be found. (3)
31. June 2019 · Paper 2 · Q11(a) · 5 marks
11. y C x Figure 8 Figure 8 shows a sketch of the curve C with equation y = x x , x < 0 (a) Find, by firstly taking logarithms, the x coordinate of the turning point of C. (Solutions based entirely on graphical or numerical methods are not acceptable.) (5)
32. 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)
33. June 2018 · Paper 1 · Q9(a) · 4 marks
9. North y Q East x P Figure 4 Figure 4 shows a sketch of the curve with equation x2 - 2xy + 3y2 = 50 (a) Show that d y y x - = d x y x - (4)
34. June 2018 · Paper 2 · Q11(b) · 3 marks
+ x - x f (x) = 1 ) x > 3 x - - x ( 3 1 )( (b) Prove that f (x) is a decreasing function. (3)
35. June 2018 · Paper 2 · Q13 · 10 marks
13. y C l P(e, e) R x Figure 2 Figure 2 shows a sketch of part of the curve C with equation y = x ln x, x > 0 The line l is the normal to C at the point P(e, e) The region R, shown shaded in Figure 2, is bounded by the curve C, the line l and the x-axis. Show that the exact area of R is Ae2 + B where A and B are rational numbers to be found. (10)
36. June 2018 · Paper 2 · Q14(b) · 4 marks
(b) Show that the rate of growth d N t is given by d N t = N N - ( ) d d 1200 (4)

Mark schemes

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

June 2025 · Paper 1 · Q9
Mark scheme June 2025 Paper 1 Q9(b)
June 2025 · Paper 2 · Q5
Mark scheme June 2025 Paper 2 Q5(c)
June 2025 · Paper 2 · Q15
Mark scheme June 2025 Paper 2 Q15(a)
June 2024 · Paper 1 · Q5
Mark scheme June 2024 Paper 1 Q5(a)
June 2024 · Paper 2 · Q6
Mark scheme June 2024 Paper 2 Q6(b)
June 2024 · Paper 2 · Q10
Mark scheme June 2024 Paper 2 Q10(a)
June 2024 · Paper 2 · Q15
Mark scheme June 2024 Paper 2 Q15(a)
June 2023 · Paper 1 · Q15
Mark scheme June 2023 Paper 1 Q15(b)
June 2023 · Paper 2 · Q7
Mark scheme June 2023 Paper 2 Q7(a)
June 2023 · Paper 2 · Q11
Mark scheme June 2023 Paper 2 Q11(a)
June 2022 · Paper 1 · Q8
Mark scheme June 2022 Paper 1 Q8(b)
June 2022 · Paper 2 · Q12
Mark scheme June 2022 Paper 2 Q12(a)
June 2022 · Paper 2 · Q16
Mark scheme June 2022 Paper 2 Q16(a)
Oct 2021 · Paper 1 · Q4
Mark scheme Oct 2021 Paper 1 Q4(a)
Oct 2021 · Paper 1 · Q14
Mark scheme Oct 2021 Paper 1 Q14
Oct 2021 · Paper 2 · Q8
Mark scheme Oct 2021 Paper 2 Q8(a)
Oct 2021 · Paper 2 · Q13
Mark scheme Oct 2021 Paper 2 Q13(a)
Oct 2021 · Paper 2 · Q14
Mark scheme Oct 2021 Paper 2 Q14(a)
Oct 2020 · Paper 1 · Q9
Mark scheme Oct 2020 Paper 1 Q9(a)
Oct 2020 · Paper 1 · Q14
Mark scheme Oct 2020 Paper 1 Q14(a)
Oct 2020 · Paper 1 · Q15
Mark scheme Oct 2020 Paper 1 Q15(b)
Oct 2020 · Paper 2 · Q7
Mark scheme Oct 2020 Paper 2 Q7(a)
Oct 2020 · Paper 2 · Q13
Mark scheme Oct 2020 Paper 2 Q13(b)
June 2019 · Paper 1 · Q3
Mark scheme June 2019 Paper 1 Q3(a)
June 2019 · Paper 1 · Q12
Mark scheme June 2019 Paper 1 Q12(a)
June 2019 · Paper 1 · Q14
Mark scheme June 2019 Paper 1 Q14(c)
June 2019 · Paper 2 · Q11
Mark scheme June 2019 Paper 2 Q11(a)
June 2018 · Paper 1 · Q5
Mark scheme June 2018 Paper 1 Q5
June 2018 · Paper 1 · Q9
Mark scheme June 2018 Paper 1 Q9(a)
June 2018 · Paper 2 · Q11
Mark scheme June 2018 Paper 2 Q11(b)
June 2018 · Paper 2 · Q13
Mark scheme June 2018 Paper 2 Q13
June 2018 · Paper 2 · Q14
Mark scheme June 2018 Paper 2 Q14(b)