: Edexcel past-paper questions pack
Back to the topic page

Exponentials and logarithms

Past-paper questions pack

Name
Class
Date set
Date due
Score/ 367

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 · 121 parts, 367 marks · mark schemes follow the questions · www.mathsaleveltutor.com · Questions and mark schemes © Pearson Education Ltd

1. June 2025 · Paper 1 · Q9(a) · 2 marks
9. v T t Figure 2 A racing car is driven along a straight road. Figure 2 shows a graph of the speed of the car as it travels along the road. The car starts from rest and is driven for T seconds before stopping. The speed of the car is modelled by the equation v = 15t – t e 0.2t 0 t T where t seconds is the time after the car starts to move. According to the model, (a) find the value of T, giving your answer to one decimal place, (2)
2. 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)
3. June 2025 · AS Paper 1 · Q10 · 5 marks
10. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Solve the equation 2 log3 (x + 1) = 1 + log3 (x + 7) (5)
4. June 2025 · Paper 1 · Q11(a) · 4 marks
11. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. The value of a car, £V, is modelled by the equation V = 1500 + A e –kt where A and k are positive constants and t is the age of the car in years. Given that • the initial value of the car was £20 000 • the value of the car was £12 000 when it was 2.5 years old (a) find a complete equation for the model, giving the exact value of A and the value of k to 3 significant figures. (4)
5. June 2025 · Paper 1 · Q11(b) · 3 marks
(b) Show that the rate of change in the value of the car can be expressed in the form –k(V – 1500) (3)
6. June 2025 · AS Paper 1 · Q13(a) · 4 marks
13. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. The growth of a particular tree is monitored over a period of time. The height, h metres, of this tree, t years after it was planted, is modelled by the equation –kt h = 31 – Ae where A and k are positive constants. Given that • exactly 10 years after it was planted, the height of the tree was 6 m • exactly 20 years after it was planted, the height of the tree was 11 m (a) find a complete equation for h in terms of t, giving the value of each of A and k to 3 significant figures. (4)
7. June 2025 · AS Paper 1 · Q13(b) · 1 marks
Use the equation of the model to answer parts (b), (c) and (d). According to the model, there is a limit to the height to which this tree can grow. (b) Deduce this limit. (1)
8. June 2025 · AS Paper 1 · Q13(d)(i) · 2 marks
(d) (i) Find d h t , giving your answer in simplest form. d (2)
9. June 2025 · AS Paper 1 · Q13(d)(ii) · 3 marks
(ii) Hence find the value of t for which the height of the tree is increasing at a rate of 30 cm per year. (3)
10. June 2025 · Paper 2 · Q3 · 2 marks
3. Given that x = 7 y find the exact value of x y (2)
11. June 2025 · Paper 2 · Q9(b) · 3 marks
Given that N = 3000 when t = T (b) find the value of T giving the answer to 2 decimal places. (3)
12. June 2025 · Paper 2 · Q9(c) · 2 marks
(c) Find the rate of increase in the total number of cars sold when t = 3, giving the answer to 3 significant figures. (2)
13. June 2025 · Paper 2 · Q10(b) · 5 marks
3 of water in the container, Given that when the tap was opened, there was 0.25 m (b) solve the differential equation to show that –kt V = P – Qe where P, Q and k are positive constants to be found. (5)
14. June 2024 · AS Paper 1 · Q7(a) · 4 marks
7. log10 P (0, 3.3) (6, 2.1) x Figure 2 A chimney emits smoke particles. On a particular day, the concentration of smoke particles in the air emitted by this chimney, P parts per million, is measured at various distances, x km, from the chimney. Figure 2 shows a sketch of the linear relationship between log10 P and x that is used to model this situation. The line passes through the point (0, 3.3) and the point (6, 2.1) (a) Find a complete equation for the model in the form P = ab x where a and b are constants. Give the value of a and the value of b each to 4 significant figures. (4)
15. June 2024 · AS Paper 1 · Q7(b) · 1 marks
(b) With reference to the model, interpret the value of ab (1)
16. June 2024 · Paper 1 · Q7(b) · 3 marks
(b) find the time taken for the depth of the water to decrease to 1.2 m. Give your answer in hours and minutes, to the nearest minute. (3)
17. June 2024 · AS Paper 1 · Q9(a) · 1 marks
9. p = loga 16 q = loga 25 where a is a constant. Find in terms of p and/or q, (a) loga 256 (1)
18. June 2024 · Paper 1 · Q9(a) · 3 marks
9. The first 3 terms of a geometric sequence are / k k k 7 2 where k is a constant. (a) Using algebra and making your reasoning clear, prove that k = (3)
19. June 2024 · AS Paper 1 · Q9(b) · 2 marks
(b) loga 100 (2)
20. June 2024 · AS Paper 1 · Q9(c) · 2 marks
(c) loga 80 x loga 3.2 (2)
21. June 2024 · AS Paper 1 · Q11(b) · 4 marks
When t = T, the rate of increase in the price per gram of metal A was equal to the rate of decrease in the price per gram of metal B (b) Find the value of T, giving your answer to one decimal place. (Solutions based entirely on calculator technology are not acceptable.) (4)
22. 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)
23. June 2024 · Paper 2 · Q12(b) · 5 marks
The volume, V microlitres, of a plant cell t hours after the plant is watered is modelled by the differential equation V d / V V t d The plant cell has an initial volume of 20 microlitres. (b) Find, according to the model, the time taken, in minutes, for the volume of the plant cell to reach 24 microlitres. (5)
24. June 2024 · Paper 2 · Q12(c) · 3 marks
(c) Show that A V kt e B where A, B and k are constants to be found. (3)
25. June 2024 · Paper 2 · Q12(d) · 2 marks
The model predicts that there is an upper limit, L microlitres, on the volume of the plant cell. (d) Find the value of L, giving a reason for your answer. (2)
26. June 2024 · Paper 2 · Q13(a) · 4 marks
13. The world human population, P billions, is modelled by the equation t P = ab where a and b are constants and t is the number of years after 2004 Using the estimated population figures for the years from 2004 to 2007, a graph is plotted of log10P against t. The points lie approximately on a straight line with • gradient 0.0054 • intercept 0.81 on the log10P axis (a) Estimate, to 3 decimal places, the value of a and the value of b. (4)
27. June 2024 · Paper 2 · Q13(b) · 2 marks
In the context of the model, (b) (i) interpret the value of the constant a, (ii) interpret the value of the constant b. (2)
28. June 2023 · Paper 1 · Q6(a) · 1 marks
6. a = log2 x b = log2(x + 8) Express in terms of a and/or b (a) log2 x (1)
29. June 2023 · Paper 1 · Q6(b) · 2 marks
(b) log2(x 2 + 8x) (2)
30. June 2023 · Paper 1 · Q6(c) · 3 marks
; (c) log2 8 :9 x Give your answer in simplest form. (3)
31. June 2023 · AS Paper 1 · Q9 · 5 marks
9. Using the laws of logarithms, solve the equation / 3 log x log x (5)
32. June 2023 · AS Paper 1 · Q11(a) · 2 marks
11. The height, h metres, of a plant, t years after it was first measured, is modelled by the equation Using the model, (a) find the height of the plant when it was first measured, (2)
33. June 2023 · Paper 1 · Q11(a) · 2 marks
11. log10 V (0, 3) (10, 2.79) t Figure 2 The value, V pounds, of a mobile phone, t months after it was bought, is modelled by t V = ab where a and b are constants. Figure 2 shows the linear relationship between log10 V and t. The line passes through the points (0, 3) and (10, 2.79) as shown. Using these points, (a) find the initial value of the phone, (2)
34. June 2023 · AS Paper 1 · Q11(b) · 3 marks
(b) show that, exactly 4 years after it was first measured, the plant was growing at approximately 15.3 cm per year. (3)
35. June 2023 · Paper 1 · Q11(b) · 3 marks
(b) find a complete equation for V in terms of t, giving the exact value of a and giving the value of b to 3 significant figures. (3)
36. June 2023 · AS Paper 1 · Q11(c) · 1 marks
According to the model, there is a limit to the height to which this plant can grow. (c) Deduce the value of this limit. (1)
37. June 2023 · Paper 2 · Q3(a) · 3 marks
3. Given that log2(x + 3) + log2(x + 10) = 2 + 2 log2 x (a) show that 3x 2 – 13x – 30 = 0 (3)
38. June 2023 · Paper 2 · Q3(b) · 2 marks
(b) (i) Write down the roots of the equation 3x 2 – 13x – 30 = 0 (ii) Hence state which of the roots in part (b)(i) is not a solution of log2(x + 3) + log2(x + 10) = 2 + 2 log2 x giving a reason for your answer. (2)
39. June 2023 · Paper 2 · Q4(b) · 3 marks
Initially, the coffee was cooling at a rate of 7.5 °C per minute. (b) Find a complete equation linking H and t, giving the value of B to 3 decimal places. (3)
40. June 2023 · Paper 2 · Q9(a) · 3 marks
9. The curve C has parametric equations x = t 2 + 6t – 16 y = 6 ln(t + 3) t > –3 (a) Show that a Cartesian equation for C is y = A ln (x + B) x > –B where A and B are integers to be found. (3)
41. June 2023 · Paper 2 · Q9(b) · 4 marks
The curve C cuts the y-axis at the point P (b) Show that the equation of the tangent to C at P can be written in the form ax + by = c ln 5 where a, b and c are integers to be found. (4)
42. June 2023 · Paper 2 · Q10(b) · 4 marks
(b) Hence find the exact value of k for which ∫ 3f ( ) d x x = - (4)
43. June 2023 · Paper 2 · Q13(c) · 5 marks
(c) Find, using algebraic integration, the exact value of ∫ 0.4 x x d (3 + x ) 0.2 giving your answer in the form a ln b + c, where a, b and c are constants to be found. (5)
44. June 2022 · Paper 1 · Q4(b) · 2 marks
(b) Hence show that =∑ 6.3 x = ln k lim δ x x δ → x 2.1 where k is a constant to be found. (2)
45. June 2022 · AS Paper 1 · Q5(a) · 4 marks
5. The mass, A kg, of algae in a small pond, is modelled by the equation A = pqt where p and q are constants and t is the number of weeks after the mass of algae was first recorded. Data recorded indicates that there is a linear relationship between t and log10 A given by the equation log10 A = 0.03t + 0.5 (a) Use this relationship to find a complete equation for the model in the form A = pqt giving the value of p and the value of q each to 4 significant figures. (4)
46. June 2022 · AS Paper 1 · Q5(c) · 3 marks
(c) Find, according to the model, (i) the mass of algae in the pond when t = 8, giving your answer to the nearest 0.5 kg, (ii) the number of weeks it takes for the mass of algae in the pond to reach 4 kg. (3)
47. June 2022 · AS Paper 1 · Q8(a) · 1 marks
8. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. The air pressure, P kg/cm2, inside a car tyre, t minutes from the instant when the tyre developed a puncture is given by the equation P = k + 1.4e–0.5t t ∈ ℝ t 0 where k is a constant. Given that the initial air pressure inside the tyre was 2.2 kg/cm2 (a) state the value of k. (1)
48. June 2022 · Paper 1 · Q8(a) · 1 marks
8. v T t Figure 2 A car stops at two sets of traffic lights. Figure 2 shows a graph of the speed of the car, v ms-1, as it travels between the two sets of traffic lights. The car takes T seconds to travel between the two sets of traffic lights. The speed of the car is modelled by the equation v = (10 - 0.4t) ln(t + 1) 0 t T where t seconds is the time after the car leaves the first set of traffic lights. According to the model, (a) find the value of T (1)
49. June 2022 · AS Paper 1 · Q8(b) · 3 marks
From the instant when the tyre developed the puncture, (b) find the time taken for the air pressure to fall to 1 kg/cm2 Give your answer in minutes to one decimal place. (3)
50. June 2022 · AS Paper 1 · Q8(c) · 2 marks
(c) Find the rate at which the air pressure in the tyre is decreasing exactly 2 minutes from the instant when the tyre developed the puncture. Give your answer in kg/cm2 per minute to 3 significant figures. (2)
51. June 2022 · AS Paper 1 · Q9(a) · 2 marks
9. (a) Given that p = log3 x, where x > 0, find in simplest form in terms of p, x (i) log3 ( ) (ii) log3 x (2)
52. June 2022 · AS Paper 1 · Q9(b) · 4 marks
(b) Hence, or otherwise, solve x ( ) = –11 2 log3 + 3 log3 x giving your answer as a simplified fraction. Solutions relying on calculator technology are not acceptable. (4)
53. June 2022 · Paper 1 · Q10(b) · 3 marks
(b) show that, exactly 10 years after the start of the study, the number of bees was increasing at a rate of approximately 18 thousand per year. (3)
54. June 2022 · Paper 1 · Q10(c) · 4 marks
The number of wasps, measured in thousands, Nw , is modelled by the equation Nw = 10 + 800 e-0.05t where t is the number of years from the start of the study. When t = T , according to the models, there are an equal number of bees and wasps. (c) Find the value of T to 2 decimal places. (4)
55. June 2022 · Paper 1 · Q12 · 5 marks
12. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Show that e : x x x a b ln d e where a and b are rational constants to be found. (5)
56. June 2022 · Paper 2 · Q2(a) · 2 marks
2. (a) Sketch the curve with equation x y = 4 stating any points of intersection with the coordinate axes. (2)
57. June 2022 · Paper 2 · Q2(b) · 2 marks
(b) Solve x = 100 giving your answer to 2 decimal places. (2)
58. June 2022 · Paper 2 · Q5(b) · 3 marks
Using your answer to part (a) and making your method clear, estimate ∫ ( ) (b) (i) 2x x log3 d ∫ (ii) 18x x log3 d (3)
59. June 2022 · Paper 2 · Q14(b) · 5 marks
When chemical A and chemical B are mixed, oxygen is produced. A scientist mixed these two chemicals and measured the total volume of oxygen produced over a period of time. 3 , t hours after the chemicals were mixed, is The total volume of oxygen produced, V m modelled by the differential equation d V V t = V 0 t k d t - t + ( )( ) where k is a constant. 3 of Given that exactly 2 hours after the chemicals were mixed, a total volume of 3 m oxygen had been produced, (b) solve the differential equation to show that t - 3 2 ( ) V = t + ( ) (5)
60. June 2022 · Paper 3 (Statistics) · Q6(c) · 5 marks
The equation of the line of best fit of y on x is y = – 0.05x + 1.92 (c) Use the equation of the line of best fit of y on x to find a model for h on m in the form h = am k where a and k are constants to be found. (5)
61. Oct 2021 · AS Paper 1 · Q2 · 3 marks
2. In this question you should show all stages of your working. Solutions relying on calculator technology are not acceptable. Given x - = y + express y in terms of x, writing your answer in simplest form. (3)
62. Oct 2021 · Paper 1 · Q5(b) · 3 marks
(b) find the first year when the yearly profit will exceed £65 000 (3)
63. Oct 2021 · Paper 1 · Q8(a) · 4 marks
8. A scientist is studying the growth of two different populations of bacteria. The number of bacteria, N, in the first population is modelled by the equation N = Aekt t 0 where A and k are positive constants and t is the time in hours from the start of the study. Given that • there were 1000 bacteria in this population at the start of the study • it took exactly 5 hours from the start of the study for this population to double (a) find a complete equation for the model. (4)
64. Oct 2021 · Paper 1 · Q8(b) · 2 marks
(b) Hence find the rate of increase in the number of bacteria in this population exactly 8 hours from the start of the study. Give your answer to 2 significant figures. (2)
65. Oct 2021 · Paper 1 · Q8(c) · 3 marks
The number of bacteria, M, in the second population is modelled by the equation M = 500e1.4kt t 0 where k has the value found in part (a) and t is the time in hours from the start of the study. Given that T hours after the start of the study, the number of bacteria in the two different populations was the same, (c) find the value of T. (3)
66. Oct 2021 · AS Paper 1 · Q11(a) · 1 marks
11. The owners of a nature reserve decided to increase the area of the reserve covered by trees. Tree planting started on 1st January 2005. The area of the nature reserve covered by trees, A km 2, is modelled by the equation A = 80 – 45e ct where c is a constant and t is the number of years after 1st January 2005. Using the model, (a) find the area of the nature reserve that was covered by trees just before tree planting started. (1)
67. Oct 2021 · AS Paper 1 · Q11(b) · 4 marks
On 1st January 2019 an area of 60 km 2 of the nature reserve was covered by trees. (b) Use this information to find a complete equation for the model, giving your value of c to 3 significant figures. (4)
68. Oct 2021 · Paper 1 · Q11(b) · 5 marks
(b) Use algebraic integration to find the exact area of R, giving your answer in the form y = a (ln 2) 2 + b ln 2 + c where a, b and c are integers to be found. (5)
69. Oct 2021 · AS Paper 1 · Q13(a) · 3 marks
13. log10 h 2.25 log10 m Figure 2 The resting heart rate, h, of a mammal, measured in beats per minute, is modelled by the equation q h = pm where p and q are constants and m is the mass of the mammal measured in kg. Figure 2 illustrates the linear relationship between log10 h and log10 m The line meets the vertical log10 h axis at 2.25 and has a gradient of –0.235 (a) Find, to 3 significant figures, the value of p and the value of q. (3)
70. Oct 2021 · AS Paper 1 · Q13(b) · 3 marks
A particular mammal has a mass of 5 kg and a resting heart rate of 119 beats per minute. (b) Comment on the suitability of the model for this mammal. (3)
71. Oct 2021 · AS Paper 1 · Q13(c) · 1 marks
(c) With reference to the model, interpret the value of the constant p. (1)
72. Oct 2021 · Paper 2 · Q3 · 3 marks
3. Using the laws of logarithms, solve the equation log3 (12y + 5) – log3 (1 – 3y) = 2 (3)
73. Oct 2021 · Paper 2 · Q10(a) · 2 marks
10. The time, T seconds, that a pendulum takes to complete one swing is modelled by the formula T = al b where l metres is the length of the pendulum and a and b are constants. (a) Show that this relationship can be written in the form log10 T = b log10 l + log10 a (2)
74. Oct 2021 · Paper 2 · Q10(b) · 3 marks
log10 T (0.21, 0.45) log10 l (– 0.7, 0) Figure 3 A student carried out an experiment to find the values of the constants a and b. The student recorded the value of T for different values of l. Figure 3 shows the linear relationship between log10 l and log10 T for the student’s data. The straight line passes through the points (– 0.7, 0) and (0.21, 0.45) Using this information, (b) find a complete equation for the model in the form T = al b giving the value of a and the value of b, each to 3 significant figures. (3)
75. Oct 2021 · Paper 2 · Q10(c) · 1 marks
(c) With reference to the model, interpret the value of the constant a. (1)
76. Oct 2021 · Paper 2 · Q14(b) · 6 marks
Given that when the tap was opened, the depth of water in the tank was 2 m, (b) show that, according to the model, h = A + B e–kt where A, B and k are constants to be found. (6)
77. Oct 2021 · Paper 3 (Statistics) · Q6(a) · 5 marks
6. The discrete random variable X has the following probability distribution x a b c P(X = x) log36 a log36 b log36 c where • a, b and c are distinct integers (a < b < c) • all the probabilities are greater than zero (a) Find (i) the value of a (ii) the value of b (iii) the value of c Show your working clearly. (5)
78. Oct 2021 · Paper 3 (Statistics) · Q6(b) · 2 marks
The independent random variables X1 and X2 each have the same distribution as X (b) Find P(X1 = X2) (2)
79. Oct 2020 · Paper 1 · Q2 · 3 marks
2. By taking logarithms of both sides, solve the equation 43 p -1 = 5210 giving the value of p to one decimal place. (3)
80. Oct 2020 · AS Paper 1 · Q3(ii) · 3 marks
(ii) Solve the equation 3x – 2 = 2 2 (3)
81. Oct 2020 · Paper 1 · Q8 · 2 marks
8. A new smartphone was released by a company. The company monitored the total number of phones sold, n, at time t days after the phone was released. The company observed that, during this time, the rate of increase of n was proportional to n Use this information to write down a suitable equation for n in terms of t. (You do not need to evaluate any unknown constants in your equation.) (2)
82. Oct 2020 · AS Paper 1 · Q8(a) · 1 marks
8. The temperature, theta °C, of a cup of tea t minutes after it was placed on a table in a room, is modelled by the equation -t theta = 18 + 65e 8 t 0 Find, according to the model, (a) the temperature of the cup of tea when it was placed on the table, (1)
83. Oct 2020 · AS Paper 1 · Q8(b) · 3 marks
(b) the value of t, to one decimal place, when the temperature of the cup of tea was 35 °C. (3)
84. Oct 2020 · AS Paper 1 · Q8(c) · 1 marks
(c) Explain why, according to this model, the temperature of the cup of tea could not fall to 15 °C. (1)
85. Oct 2020 · AS Paper 1 · Q8(d) · 4 marks
μ (0, 94) (8, 50) l t Figure 2 The temperature, μ °C, of a second cup of tea t minutes after it was placed on a table in a different room, is modelled by the equation -t μ = A + Be 8 t 0 where A and B are constants. Figure 2 shows a sketch of μ against t with two data points that lie on the curve. The line l, also shown on Figure 2, is the asymptote to the curve. Using the equation of this model and the information given in Figure 2 (d) find an equation for the asymptote l. (4)
86. Oct 2020 · Paper 1 · Q9(b) · 3 marks
(b) Hence find, in simplest form, the exact coordinates of the stationary points of C. (3)
87. Oct 2020 · Paper 1 · Q10(b) · 6 marks
(b) Hence, using algebraic integration, show that ∫ dx = a ( ) ln ( ) x - + x - 1 3 where a is a rational constant to be found. (6)
88. Oct 2020 · AS Paper 1 · Q12(a) · 4 marks
12. An advertising agency is monitoring the number of views of an online advert. The equation log10 V = 0.072t + 2.379 1 t 30, t ∈ is used to model the total number of views of the advert, V, in the first t days after the advert went live. (a) Show that V = abt where a and b are constants to be found. Give the value of a to the nearest whole number and give the value of b to 3 significant figures. (4)
89. Oct 2020 · AS Paper 1 · Q12(b) · 1 marks
(b) Interpret, with reference to the model, the value of ab. (1)
90. Oct 2020 · AS Paper 1 · Q12(c) · 2 marks
Using this model, calculate (c) the total number of views of the advert in the first 20 days after the advert went live. Give your answer to 2 significant figures. (2)
91. Oct 2020 · Paper 2 · Q3(a) · 3 marks
3. (a) Given that 2 log (4 - x) = log (x + 8) show that x2 - 9x + 8 = 0 (3)
92. Oct 2020 · Paper 2 · Q3(b) · 2 marks
(b) (i) Write down the roots of the equation x2 - 9x + 8 = 0 (ii) State which of the roots in (b)(i) is not a solution of 2 log (4 - x) = log (x + 8) giving a reason for your answer. (2)
93. Oct 2020 · Paper 2 · Q5 · 4 marks
x+1 at the point P. x meets the curve with equation y = 15 - 2 5. The curve with equation y = 3 x 2 Find, using algebra, the exact x coordinate of P. (4)
94. Oct 2020 · Paper 2 · Q6(b) · 4 marks
(b) Hence, using algebraic integration, find the exact value of ∫ x + x - x d x + giving your answer in the form a + b ln 2 where a and b are integers to be found. (4)
95. Oct 2020 · Paper 2 · Q9(a) · 4 marks
9. A quantity of ethanol was heated until it reached boiling point. The temperature of the ethanol, theta °C, at time t seconds after heating began, is modelled by the equation theta = A - Be- 0.07t where A and B are positive constants. Given that ● the initial temperature of the ethanol was 18 °C ● after 10 seconds the temperature of the ethanol was 44 °C (a) find a complete equation for the model, giving the values of A and B to 3 significant figures. (4)
96. Oct 2020 · Paper 2 · Q13(a) · 1 marks
13. The function g is defined by x - ln( ) g (x) = x > 0 x ≠ k x - ln( ) where k is a constant. (a) Deduce the value of k. (1)
97. Oct 2020 · Paper 2 · Q13(c) · 2 marks
(c) Find the range of values of a for which g (a) > 0 (2)
98. June 2019 · Paper 1 · Q7(a) · 4 marks
7. In a simple model, the value, £V, of a car depends on its age, t, in years. The following information is available for car A Ɣ its value when new is £20 000 Ɣ its value after one year is £16 000 (a) Use an exponential model to form, for car A, a possible equation linking V with t. (4)
99. June 2019 · Paper 1 · Q9(a) · 3 marks
9. Given that a < b < 0 and that a and b satisfy the equation log a – log b = log(a – b) (a) show that b a = b - (3)
100. June 2019 · Paper 1 · Q9(b) · 2 marks
(b) Write down the full restriction on the value of b, explaining the reason for this restriction. (2)
101. June 2019 · Paper 1 · Q12(b) · 2 marks
y x Figure 3 Figure 3 shows a sketch of part of the curve with equation y = f(x). (b) Sketch the graph of H against t where H(t) = Ň10e±௘0.25t sin tŇ = = t . 0 showing the long-term behaviour of this curve. (2)
102. June 2019 · Paper 1 · Q13(b) · 8 marks
y C R x Figure 4 Figure 4 shows a sketch of part of the curve C. The region R, shown shaded in Figure 4, is bounded by the curve C, the x-axis and the line with equation x = 3 (b) Show that the exact value of the area of R is a ln 2 + b ln 3, where a and b are rational constants to be found. (8)
103. June 2019 · AS Paper 1 · Q14(b) · 6 marks
Given the model predicts that the value of the car is decreasing at a rate of £500 per year at the instant when t = T, (b) (i) show that 3925e– 0.25T = 500 (ii) Hence find the age of the car at this instant, giving your answer in years and months to the nearest month. (Solutions based entirely on graphical or numerical methods are not acceptable.) (6)
104. June 2019 · AS Paper 1 · Q14(c) · 1 marks
The model predicts that the value of the car approaches, but does not fall below, £A. (c) State the value of A. (1)
105. June 2019 · Paper 2 · Q1 · 3 marks
1. Given x y x = 2 2 express y as a function of x. (3)
106. June 2019 · Paper 2 · Q8(ii) · 3 marks
(ii) Show that =∑ n + ⎛ ⎞ log5 ⎠⎟= ⎝⎜ n + n (3)
107. June 2019 · Paper 2 · Q9(a) · 3 marks
9. A research engineer is testing the effectiveness of the braking system of a car when it is driven in wet conditions. The engineer measures and records the braking distance, d metres, when the brakes are applied from a speed of V ௘km hí1 . Graphs of d against V and log10 d against log10 V were plotted. The results are shown below together with a data point from each graph. log10 d d (30, 20) log10 V (0, í1.77) V Figure 5 Figure 6 (a) Explain how Figure 6 would lead the engineer to believe that the braking distance should be modelled by the formula d = kV n where k and n are constants with k | 0.017 (3)
108. June 2019 · Paper 2 · Q9(b) · 3 marks
Using the information given in Figure 5, with k = 0.017 (b) find a complete equation for the model giving the value of n to 3 significant figures. (3)
109. June 2019 · Paper 2 · Q9(c) · 3 marks
Sean is driving this car at 60 km hí1 in wet conditions when he notices a large puddle in the road 100 m ahead. It takes him 0.8 seconds to react before applying the brakes. (c) Use your formula to find out if Sean will be able to stop before reaching the puddle. (3)
110. 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)
111. June 2019 · Paper 3 (Statistics) · Q3(c) · 5 marks
(c) Show that the relationship between y and x can be written in the form y = ax” where a and v are constants to be found. (5)
112. June 2018 · Paper 1 · Q4(a) · 2 marks
4. The curve with equation y = 2 ln(8 - x) meets the line y = x at a single point, x = alpha . (a) Show that 3 < alpha < 4 (2)
113. June 2018 · AS Paper 1 · Q5(a) · 2 marks
5. A student’s attempt to solve the equation 2 log2 x í log2 x = 3 is shown below. 2 log2 x í log2 x = 3 x ⎛ ⎞ 2 log2 ⎠⎟ = 3 using the subtraction law for logs ⎝⎜ x ( ) = 3 2 log2 x simplifying log2 x = 3 using the power law for logs x = 32 = 9 using the definition of a log (a) Identify two errors made by this student, giving a brief explanation of each. (2)
114. June 2018 · AS Paper 1 · Q5(b) · 3 marks
(b) Write out the correct solution. (3)
115. June 2018 · Paper 1 · Q7(a) · 4 marks
7. Given that k + k ∫ (a) show that d is independent of k, x ( x - k ) (4) k
116. June 2018 · Paper 1 · Q10(a) · 5 marks
10. The height above ground, H metres, of a passenger on a roller coaster can be modelled by the differential equation d H H cos( . 0 25 t ) = d t where t is the time, in seconds, from the start of the ride. Given that the passenger is 5 m above the ground at the start of the ride, (a) show that H = 5e0.1 sin (0.25t) (5)
117. June 2018 · Paper 1 · Q12(a) · 4 marks
12. The value, £V, of a vintage car t years after it was first valued on 1st January 2001, is modelled by the equation V = Ap t where A and p are constants Given that the value of the car was £32 000 on 1st January 2005 and £50 000 on 1st January 2012 (a) (i) find p to 4 decimal places, (ii) show that A is approximately 24 800 (4)
118. June 2018 · Paper 1 · Q12(c) · 4 marks
Using the model, (c) find the year during which the value of the car first exceeds £100 000 (4)
119. June 2018 · AS Paper 1 · Q13(a) · 4 marks
13. log10V l t Figure 3 The value of a rare painting, £V, is modelled by the equation V = pq௘t, where p and q are constants and t is the number of years since the value of the painting was first recorded on 1st January 1980. The line l shown in Figure 3 illustrates the linear relationship between t and log10V since 1st January 1980. The equation of line l is log10V = 0.05t + 4.8 (a) Find, to 4 significant figures, the value of p and the value of q. (4)
120. June 2018 · AS Paper 1 · Q13(b) · 2 marks
(b) With reference to the model interpret (i) the value of the constant p, (ii) the value of the constant q. (2)
121. June 2018 · Paper 2 · Q14(c) · 4 marks
The rate of growth is a maximum after T months. (c) Find, according to the model, the value of T. (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 · AS Paper 1 · Q10
Mark scheme June 2025 Paper 1 Q10
June 2025 · Paper 1 · Q11
Mark scheme June 2025 Paper 1 Q11(b)
June 2025 · AS Paper 1 · Q13
Mark scheme June 2025 Paper 1 Q13(d)(ii)
June 2025 · Paper 2 · Q3
Mark scheme June 2025 Paper 2 Q3
June 2025 · Paper 2 · Q9
Mark scheme June 2025 Paper 2 Q9(c)
June 2025 · Paper 2 · Q10
Mark scheme June 2025 Paper 2 Q10(b)
June 2024 · AS Paper 1 · Q7
Mark scheme June 2024 Paper 1 Q7(b)
June 2024 · Paper 1 · Q7
Mark scheme June 2024 Paper 1 Q7(b)
June 2024 · AS Paper 1 · Q9
Mark scheme June 2024 Paper 1 Q9(c)
June 2024 · Paper 1 · Q9
Mark scheme June 2024 Paper 1 Q9(a)
June 2024 · AS Paper 1 · Q11
Mark scheme June 2024 Paper 1 Q11(b)
June 2024 · Paper 2 · Q6
Mark scheme June 2024 Paper 2 Q6(b)
June 2024 · Paper 2 · Q12
Mark scheme June 2024 Paper 2 Q12(d)
June 2024 · Paper 2 · Q13
Mark scheme June 2024 Paper 2 Q13(b)
June 2023 · Paper 1 · Q6
Mark scheme June 2023 Paper 1 Q6(c)
June 2023 · AS Paper 1 · Q9
Mark scheme June 2023 Paper 1 Q9
June 2023 · AS Paper 1 · Q11
Mark scheme June 2023 Paper 1 Q11(c)
June 2023 · Paper 1 · Q11
Mark scheme June 2023 Paper 1 Q11(b)
June 2023 · Paper 2 · Q3
Mark scheme June 2023 Paper 2 Q3(b)
June 2023 · Paper 2 · Q4
Mark scheme June 2023 Paper 2 Q4(b)
June 2023 · Paper 2 · Q9
Mark scheme June 2023 Paper 2 Q9(b)
June 2023 · Paper 2 · Q10
Mark scheme June 2023 Paper 2 Q10(b)
June 2023 · Paper 2 · Q13
Mark scheme June 2023 Paper 2 Q13(c)
June 2022 · Paper 1 · Q4
Mark scheme June 2022 Paper 1 Q4(b)
June 2022 · AS Paper 1 · Q5
Mark scheme June 2022 Paper 1 Q5(c)
June 2022 · AS Paper 1 · Q8
Mark scheme June 2022 Paper 1 Q8(c)
June 2022 · Paper 1 · Q8
Mark scheme June 2022 Paper 1 Q8(a)
June 2022 · AS Paper 1 · Q9
Mark scheme June 2022 Paper 1 Q9(b)
June 2022 · Paper 1 · Q10
Mark scheme June 2022 Paper 1 Q10(c)
June 2022 · Paper 1 · Q12
Mark scheme June 2022 Paper 1 Q12
June 2022 · Paper 2 · Q2
Mark scheme June 2022 Paper 2 Q2(b)
June 2022 · Paper 2 · Q5
Mark scheme June 2022 Paper 2 Q5(b)
June 2022 · Paper 2 · Q14
Mark scheme June 2022 Paper 2 Q14(b)
June 2022 · Paper 3 (Statistics) · Q6
Mark scheme June 2022 Paper 3 (Statistics) Q6(c)
Oct 2021 · AS Paper 1 · Q2
Mark scheme Oct 2021 Paper 1 Q2
Oct 2021 · Paper 1 · Q5
Mark scheme Oct 2021 Paper 1 Q5(b)
Oct 2021 · Paper 1 · Q8
Mark scheme Oct 2021 Paper 1 Q8(c)
Oct 2021 · AS Paper 1 · Q11
Mark scheme Oct 2021 Paper 1 Q11(b)
Oct 2021 · Paper 1 · Q11
Mark scheme Oct 2021 Paper 1 Q11(b)
Oct 2021 · AS Paper 1 · Q13
Mark scheme Oct 2021 Paper 1 Q13(c)
Oct 2021 · Paper 2 · Q3
Mark scheme Oct 2021 Paper 2 Q3
Oct 2021 · Paper 2 · Q10
Mark scheme Oct 2021 Paper 2 Q10(c)
Oct 2021 · Paper 2 · Q14
Mark scheme Oct 2021 Paper 2 Q14(b)
Oct 2021 · Paper 3 (Statistics) · Q6
Mark scheme Oct 2021 Paper 3 (Statistics) Q6(b)
Oct 2020 · Paper 1 · Q2
Mark scheme Oct 2020 Paper 1 Q2
Oct 2020 · AS Paper 1 · Q3
Mark scheme Oct 2020 Paper 1 Q3(ii)
Oct 2020 · Paper 1 · Q8
Mark scheme Oct 2020 Paper 1 Q8
Oct 2020 · AS Paper 1 · Q8
Mark scheme Oct 2020 Paper 1 Q8(d)
Oct 2020 · Paper 1 · Q9
Mark scheme Oct 2020 Paper 1 Q9(b)
Oct 2020 · Paper 1 · Q10
Mark scheme Oct 2020 Paper 1 Q10(b)
Oct 2020 · AS Paper 1 · Q12
Mark scheme Oct 2020 Paper 1 Q12(c)
Oct 2020 · Paper 2 · Q3
Mark scheme Oct 2020 Paper 2 Q3(b)
Oct 2020 · Paper 2 · Q5
Mark scheme Oct 2020 Paper 2 Q5
Oct 2020 · Paper 2 · Q6
Mark scheme Oct 2020 Paper 2 Q6(b)
Oct 2020 · Paper 2 · Q9
Mark scheme Oct 2020 Paper 2 Q9(a)
Oct 2020 · Paper 2 · Q13
Mark scheme Oct 2020 Paper 2 Q13(c)
June 2019 · Paper 1 · Q7
Mark scheme June 2019 Paper 1 Q7(a)
June 2019 · Paper 1 · Q9
Mark scheme June 2019 Paper 1 Q9(b)
June 2019 · Paper 1 · Q12
Mark scheme June 2019 Paper 1 Q12(b)
June 2019 · Paper 1 · Q13
Mark scheme June 2019 Paper 1 Q13(b)
June 2019 · AS Paper 1 · Q14
Mark scheme June 2019 Paper 1 Q14(c)
June 2019 · Paper 2 · Q1
Mark scheme June 2019 Paper 2 Q1
June 2019 · Paper 2 · Q8
Mark scheme June 2019 Paper 2 Q8(ii)
June 2019 · Paper 2 · Q9
Mark scheme June 2019 Paper 2 Q9(c)
June 2019 · Paper 2 · Q11
Mark scheme June 2019 Paper 2 Q11(a)
June 2019 · Paper 3 (Statistics) · Q3
Mark scheme June 2019 Paper 3 (Statistics) Q3(c)
June 2018 · Paper 1 · Q4
Mark scheme June 2018 Paper 1 Q4(a)
June 2018 · AS Paper 1 · Q5
Mark scheme June 2018 Paper 1 Q5(b)
June 2018 · Paper 1 · Q7
Mark scheme June 2018 Paper 1 Q7(a)
June 2018 · Paper 1 · Q10
Mark scheme June 2018 Paper 1 Q10(a)
June 2018 · Paper 1 · Q12
Mark scheme June 2018 Paper 1 Q12(c)
June 2018 · AS Paper 1 · Q13
Mark scheme June 2018 Paper 1 Q13(b)
June 2018 · Paper 2 · Q14
Mark scheme June 2018 Paper 2 Q14(c)