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Differentiation: first principles, gradients, stationary points

Past-paper questions pack

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Class
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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 · 140 parts, 449 marks · mark schemes follow the questions · www.mathsaleveltutor.com · Questions and mark schemes © Pearson Education Ltd

1. June 2025 · AS Paper 1 · Q6(b) · 5 marks
Given also that f′ (2) = 14 (b) find the value of a and the value of b. (5)
2. June 2025 · Paper 1 · Q7(a) · 2 marks
7. y –1 x –75 C Figure 1 Figure 1 shows a sketch of a curve C with equation y = f(x), where f(x) is a quartic expression in x. The curve • has maximum turning points at (–1, 0) and (5, 0) • crosses the y‑axis at (0, –75) • has a minimum turning point at x = 2 (a) Find the set of values of x for which f′(x) 0 writing your answer in set notation. (2)
3. June 2025 · Paper 1 · Q7(c) · 2 marks
The curve C1 has equation y = f(x) + k, where k is a constant. Given that the graph of C1 intersects the x‑axis at exactly four places, (c) find the range of possible values for k. (2)
4. 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)
5. June 2025 · AS 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. l cm r cm Figure 4 Figure 4 shows a design for a feeding trough. The trough is modelled as a hollow, semicircular cylinder of radius r cm and length l cm. The trough will be made from sheet metal of negligible thickness. Given that the capacity of the trough will be 90 000 pi cm 2, of sheet metal required to make the trough is (a) show that the total area, A cm given by A = 180000 pi + pi r r (4)
6. June 2025 · AS Paper 1 · Q11(b) · 4 marks
(b) Use calculus to find the radius of the trough for which A is a minimum. (4)
7. 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)
8. June 2025 · AS Paper 1 · Q11(c) · 2 marks
(c) Show that the radius found in part (b) gives the minimum value of A. (2)
9. June 2025 · AS Paper 1 · Q11(d) · 2 marks
Given that the sheet metal costs £30 per square metre (d) calculate the minimum cost of sheet metal required to make one trough. (2)
10. June 2025 · AS Paper 1 · Q11(e) · 1 marks
(e) State one assumption you have made in calculating your answer to part (d). (1)
11. June 2025 · AS Paper 1 · Q12(a) · 3 marks
12. y C x Figure 5 Figure 5 shows a sketch of the curve C. The curve has a minimum turning point on the y-axis and a maximum turning point in quadrant 2. (a) On Diagram 1, sketch the graph of the gradient function for C. [A copy of Figure 5 has been included on the next page to help you.] (3)
12. June 2025 · AS Paper 1 · Q12(b) · 2 marks
The gradient function for C has one of the following equations y y y y d d d d x x x O 3 x x x x ( )1 ( ) x x x x d d d d (b) State the correct equation for the gradient function, fully justifying your answer. (2)
13. June 2025 · AS Paper 1 · Q13(d)(i) · 2 marks
(d) (i) Find d h t , giving your answer in simplest form. d (2)
14. June 2025 · Paper 1 · Q15(b) · 3 marks
Using algebraic differentiation, (b) find the value of r for which P has a stationary value. (3)
15. June 2025 · Paper 1 · Q15(c) · 2 marks
(c) Prove, by further differentiation, that this value of r gives the minimum perimeter of the stage. (2)
16. June 2025 · Paper 2 · Q8(a) · 4 marks
8. y C x Figure 2 . / O f x x x x ax where a is a constant. Figure 2 shows a sketch of the curve C with equation y = f (x) Given that C has a local maximum at x = – 1 (a) show that a = –4 (4)
17. June 2025 · Paper 2 · Q8(b) · 1 marks
(b) find the exact y coordinate of the local maximum. (1)
18. June 2025 · Paper 2 · Q8(c) · 4 marks
The equation f (x) = k, where k is a constant, has 4 distinct solutions. (c) Using algebra and showing all stages of your working, find the range of values of k. Give the answer using set notation. (Solutions relying on calculator technology are not acceptable.) (4)
19. June 2025 · Paper 2 · Q11(d) · 1 marks
(d) Hence, or otherwise, deduce the selling price of the toy that maximises the profit. (1)
20. 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)
21. June 2025 · AS Paper 2 (Mechanics) · Q4(a) · 4 marks
4. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. A fixed point O lies on a straight line. A particle P moves along the straight line. At time t seconds, t 0, the distance, s metres, of P from O is given by 3 – 12t 2 + 18t s = 2t (a) Find the values of t for which P is instantaneously at rest. (4)
22. June 2025 · AS Paper 2 (Mechanics) · Q4(c) · 3 marks
(c) Find the maximum speed of P in the interval 1 t 3 (3)
23. June 2025 · Paper 3 (Mechanics) · Q4(c) · 2 marks
(c) Find an expression for the velocity of P at time t seconds, where t > 0, giving your answer in terms of t, i and j (2)
24. June 2025 · Paper 3 (Mechanics) · Q4(d) · 4 marks
At t = T, the acceleration of P is in a direction that is perpendicular to the line with equation y = 3 x (d) Find the value of T. (4)
25. June 2024 · Paper 1 · Q3(b) · 2 marks
(b) Find f′(x) (2)
26. June 2024 · Paper 1 · Q4 · 3 marks
2, use differentiation from first principles to show that d y 4. Given that y = x x = 2x d (3)
27. June 2024 · Paper 1 · Q5(b) · 3 marks
(b) Hence, using algebra, find the values of x for which f is decreasing. You must show each step in your working. (3)
28. June 2024 · AS Paper 1 · Q8(a) · 3 marks
8. y C R B A x l Figure 3 In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Figure 3 shows a sketch of the curve C with equation 3 – 14x + 23 y = x The line l is the tangent to C at the point A, also shown in Figure 3. Given that l has equation y = –2x + 7 (a) show, using calculus, that the x coordinate of A is 2 (3)
29. June 2024 · Paper 1 · Q10(b) · 3 marks
The line l1 is the tangent to the curve at A. (b) Use calculus to show that an equation of line l1 is 12x + y = 48 (3)
30. June 2024 · AS Paper 1 · Q12(b) · 4 marks
(b) Use calculus to find the value of x for which P has a stationary value. (4)
31. June 2024 · AS Paper 1 · Q12(c) · 2 marks
(c) Prove, by further calculus, that this value of x gives a minimum value for P (2)
32. June 2024 · AS Paper 1 · Q12(d) · 2 marks
Access to the pool is by side AB shown in Figure 5. Given that AB must be at least one metre, (d) determine, according to the model, whether the swimming pool with the minimum perimeter would be suitable. (2)
33. June 2024 · Paper 2 · Q1(a) · 3 marks
1. y x x x (a) Find in simplest form (i) d y d x y (ii) d x d (3)
34. June 2024 · Paper 2 · Q1(b) · 2 marks
y (b) Hence find the exact value of x when d x = 0 d (2)
35. 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)
36. June 2024 · Paper 2 · Q9(a) · 4 marks
9. H (m) A(0, 2) B(20, 0.8) x (m) Figure 3 The graph in Figure 3 shows the path of a small ball. The ball travels in a vertical plane above horizontal ground. The ball is thrown from the point represented by A and caught at the point represented by B. The height, H metres, of the ball above the ground has been plotted against the horizontal distance, x metres, measured from the point where the ball was thrown. With respect to a fixed origin O, the point A has coordinates (0, 2) and the point B has coordinates (20, 0.8), as shown in Figure 3. The ball reaches its maximum height when x = 9 A quadratic function, linking H with x, is used to model the path of the ball. (a) Find H in terms of x. (4)
37. June 2024 · Paper 2 · Q15(b) · 2 marks
The point P(1, 0) lies on C. (b) Show that the normal to C at P has equation y x O 3 (2)
38. June 2024 · AS Paper 2 (Mechanics) · Q2(a) · 3 marks
2. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. A particle is moving along a straight line. At time t seconds, t > 0, the velocity of the particle is vms-1, where v t t (a) Find the acceleration of the particle when t = 4 (3)
39. June 2024 · Paper 3 (Mechanics) · Q4(b) · 4 marks
(b) Find the speed of P when t = 4 (4)
40. June 2024 · Paper 3 (Mechanics) · Q4(c) · 4 marks
When t = T, P is accelerating in the direction of (–i – 27j). (c) Find the value of T. (4)
41. June 2024 · Paper 3 (Mechanics) · Q5(c) · 2 marks
Using the model, the greatest height of the stone above the ground is found to be H metres. (c) Use the answer to (a), or otherwise, to find the value of H. (2)
42. June 2023 · AS Paper 1 · Q1(a) · 2 marks
1. A curve has equation (a) Find d y writing your answer in simplest form. d x (2)
43. June 2023 · AS Paper 1 · Q1(b) · 4 marks
(b) Hence find the range of values of x for which y is decreasing. (4)
44. June 2023 · Paper 1 · Q4(a) · 3 marks
4. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. The curve C has equation y = f(x) where x ∈ ℝ Given that f′(x) = 2x + 1 • 2 cos x • the curve has a stationary point with x coordinate alpha • alpha is small (a) use the small angle approximation for cos x to estimate the value of alpha to 3 decimal places. (3)
45. June 2023 · Paper 1 · Q4(b) · 2 marks
The point P(0, 3) lies on C (b) Find the equation of the tangent to the curve at P, giving your answer in the form y = mx + c, where m and c are constants to be found. (2)
46. 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)
47. 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)
48. 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)
49. 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)
50. June 2023 · AS Paper 1 · Q16 · 6 marks
16. A curve has equation y = f ( ), x x 0 Given that • f ( ) ∑ x x a x b , where a and b are constants / • the curve has a stationary point at 4 3 , • the curve meets the y-axis at -5 find f ( ), x giving your answer in simplest form. (6)
51. June 2023 · Paper 2 · Q1(a) · 2 marks
1. f(x) = x 3 + 2x 2 – 8x + 5 (a) Find f ′′(x) (2)
52. June 2023 · Paper 2 · Q1(b) · 2 marks
(b) (i) Solve f ′′(x) = 0 (ii) Hence find the range of values of x for which f(x) is concave. (2)
53. 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)
54. June 2023 · Paper 2 · Q5(a) · 2 marks
5. The curve C has equation y = f (x) The curve • passes through the point P(3, –10) • has a turning point at P Given that y d x x x k x d where k is a constant, (a) show that k = 12 (2)
55. June 2023 · Paper 2 · Q5(b) · 3 marks
(b) Hence find the coordinates of the point where C crosses the y-axis. (3)
56. June 2023 · Paper 2 · Q7(b) · 3 marks
The point P(–2, 5) lies on the curve. (b) Find the equation of the normal to the curve at P, giving your answer in the form ax + by + c = 0 , where a, b and c are integers to be found. (3)
57. 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)
58. June 2023 · AS Paper 2 (Mechanics) · Q3(b) · 3 marks
(b) Find the magnitude of the acceleration of P when t = 3 (3)
59. June 2023 · Paper 3 (Mechanics) · Q3(c) · 2 marks
(c) the acceleration of P at time t seconds (2)
60. June 2022 · Paper 1 · Q6(a) · 1 marks
6. y C (2, 8) x Figure 1 Figure 1 shows a sketch of a curve C with equation y = f (x) where f (x) is a cubic expression in x. The curve • passes through the origin • has a maximum turning point at (2, 8) • has a minimum turning point at (6, 0) (a) Write down the set of values of x for which f ʹ(x) < 0 (1)
61. June 2022 · AS Paper 1 · Q7(c) · 3 marks
The line l has equation y = k where k is a constant. Given that C and l intersect at 3 distinct points, (c) find the range of values for k, writing your answer in set notation. Solutions relying on calculator technology are not acceptable. (3)
62. June 2022 · Paper 1 · Q8(b) · 4 marks
(b) show that the maximum speed of the car occurs when t = - 1 + ln( t + ) (4)
63. 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)
64. June 2022 · AS Paper 1 · Q10(a) · 5 marks
10. y l P C R x Figure 2 In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Figure 2 shows a sketch of part of the curve C with equation y = 1 3 x2 – 2 x + 3 x 0 The point P lies on C and has x coordinate 4 The line l is the tangent to C at P. (a) Show that l has equation 13x – 6y – 26 = 0 (5)
65. June 2022 · AS Paper 1 · Q12(b) · 4 marks
(b) Use calculus to find the value of r for which C is a minimum, giving your answer to 3 significant figures. (4)
66. June 2022 · AS Paper 1 · Q12(c) · 2 marks
C (c) Using d prove that the cost is minimised for the value of r found in part (b). r d (2)
67. June 2022 · AS Paper 1 · Q12(d) · 2 marks
(d) Hence find the minimum value of C, giving your answer to the nearest integer. (2)
68. June 2022 · Paper 1 · Q15(b) · 4 marks
Using algebraic differentiation, (b) find the value of r for which S has a stationary point. (4)
69. June 2022 · Paper 1 · Q15(c) · 2 marks
(c) Prove, by further differentiation, that this value of r gives the minimum surface area of the toy. (2)
70. 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)
71. June 2022 · Paper 2 · Q4 · 3 marks
4. Given that y = 2x use differentiation from first principles to show that d y x = 4x d (3)
72. June 2022 · Paper 2 · Q6(a) · 4 marks
6. y alpha x P Figure 2 Figure 2 shows a sketch of part of the curve with equation y = f(x) where f(x) = 8 sin 1 2 x - 3x + 9 x > 0 and x is measured in radians. The point P, shown in Figure 2, is a local maximum point on the curve. Using calculus and the sketch in Figure 2, (a) find the x coordinate of P, giving your answer to 3 significant figures. (4)
73. June 2022 · Paper 2 · Q6(c) · 2 marks
(c) Taking x0 = 5 as a first approximation to alpha , apply the Newton-Raphson method once to f(x) to obtain a second approximation to alpha . Show your method and give your answer to 3 significant figures. (2)
74. 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)
75. June 2022 · Paper 2 · Q12(b) · 3 marks
Given that the curve with equation y = f(x) has at least one stationary point, (b) find the range of possible values of k. (3)
76. June 2022 · AS Paper 2 (Mechanics) · Q3(a) · 6 marks
3. A fixed point O lies on a straight line. A particle P moves along the straight line. At time t seconds, t 0, the distance, s metres, of P from O is given by s = 1 3 - 5 2 + 6t 3 t 2 t (a) Find the acceleration of P at each of the times when P is at instantaneous rest. (6)
77. June 2022 · Paper 3 (Mechanics) · Q1(b) · 2 marks
(b) Find an expression, in terms of t, i and j, for the acceleration of P at time t seconds, where t > 0 (2)
78. 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)
79. Oct 2021 · AS Paper 1 · Q5(a) · 2 marks
5. y Q P(2, 10) x Figure 1 Figure 1 shows part of the curve with equation y = 3x 2 – 2 The point P(2, 10) lies on the curve. (a) Find the gradient of the tangent to the curve at P. (2)
80. Oct 2021 · AS Paper 1 · Q5(b) · 3 marks
The point Q with x coordinate 2 + h also lies on the curve. (b) Find the gradient of the line PQ, giving your answer in terms of h in simplest form. (3)
81. Oct 2021 · AS Paper 1 · Q5(c) · 1 marks
(c) Explain briefly the relationship between part (b) and the answer to part (a). (1)
82. 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)
83. Oct 2021 · Paper 1 · Q12(a) · 5 marks
12. H x Figure 3 Figure 3 is a graph of the trajectory of a golf ball after the ball has been hit until it first hits the ground. The vertical height, H metres, of the ball above the ground has been plotted against the horizontal distance travelled, x metres, measured from where the ball was hit. The ball is modelled as a particle travelling in a vertical plane above horizontal ground. Given that the ball • is hit from a point on the top of a platform of vertical height 3 m above the ground • reaches its maximum vertical height after travelling a horizontal distance of 90 m • is at a vertical height of 27 m above the ground after travelling a horizontal distance of 120 m Given also that H is modelled as a quadratic function in x (a) find H in terms of x (5)
84. 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)
85. Oct 2021 · AS Paper 1 · Q14(b) · 2 marks
The curve C has a maximum turning point at M. (b) Find the coordinates of M. (2)
86. Oct 2021 · AS Paper 1 · Q16(a) · 4 marks
16. The curve C has equation y = f(x) where f(x) = ax 3 + 15x 2 – 39x + b and a and b are constants. Given • the point (2, 10) lies on C • the gradient of the curve at (2, 10) is –3 (a) (i) show that the value of a is –2 (ii) find the value of b. (4)
87. Oct 2021 · AS Paper 1 · Q16(b) · 3 marks
(b) Hence show that C has no stationary points. (3)
88. Oct 2021 · Paper 2 · Q5(a) · 3 marks
5. The curve C has equation y = 5x4 – 24x3 + 42x2 – 32x + 11 x ∈ (a) Find (i) d y d x y (ii) d x d (3)
89. Oct 2021 · Paper 2 · Q5(b) · 4 marks
(b) (i) Verify that C has a stationary point at x = 1 (ii) Show that this stationary point is a point of inflection, giving reasons for your answer. (4)
90. Oct 2021 · Paper 2 · Q7(a) · 4 marks
7. In this question you should show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. y C x R P(5, –13) l Figure 2 Figure 2 shows a sketch of part of the curve C with equation y = x3 – 10x2 + 27x – 23 The point P(5, –13) lies on C The line l is the tangent to C at P (a) Use differentiation to find the equation of l, giving your answer in the form y = mx + c where m and c are integers to be found. (4)
91. Oct 2021 · Paper 2 · Q8(b) · 5 marks
Given that • the point P (–1, – 4) lies on C • the normal to C at P has equation 19x + 26y + 123 = 0 (b) find the value of p and the value of q. (5)
92. 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)
93. Oct 2021 · AS Paper 2 (Mechanics) · Q2(a) · 2 marks
2. A particle P moves along a straight line. At time t seconds, the velocity v m s-1 of P is modelled as v = 10t - t 2 - k t 0 where k is a constant. (a) Find the acceleration of P at time t seconds. (2)
94. Oct 2021 · Paper 3 (Mechanics) · Q5(a) · 2 marks
5. At time t seconds, a particle P has velocity v m s-1, where 2 i - 2t j t > 0 v = 3t (a) Find the acceleration of P at time t seconds, where t > 0 (2)
95. Oct 2020 · AS Paper 1 · Q1 · 5 marks
1. A curve has equation y = 2x3 – 4x + 5 Find the equation of the tangent to the curve at the point P(2, 13). Write your answer in the form y = mx + c, where m and c are integers to be found. Solutions relying on calculator technology are not acceptable. (5)
96. 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)
97. Oct 2020 · Paper 1 · Q9(b) · 3 marks
(b) Hence find, in simplest form, the exact coordinates of the stationary points of C. (3)
98. Oct 2020 · Paper 1 · Q9(c) · 3 marks
The function g and the function h are defined by g (x) = 2f (x) x h (x) = 2f (x) - 3 x 0 (c) Find (i) the range of g (ii) the range of h (3)
99. Oct 2020 · AS Paper 1 · Q14(a) · 7 marks
14. A curve has equation y = g(x). Given that ● g(x) is a cubic expression in which the coefficient of x3 is equal to the coefficient of x ● the curve with equation y = g(x) passes through the origin ● the curve with equation y = g(x) has a stationary point at (2, 9) (a) find g(x), (7)
100. Oct 2020 · AS Paper 1 · Q14(b) · 2 marks
(b) prove that the stationary point at (2, 9) is a maximum. (2)
101. Oct 2020 · Paper 1 · Q15(b) · 3 marks
(b) Prove that C has a point of inflection at x = (3)
102. 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)
103. Oct 2020 · Paper 2 · Q7(b) · 3 marks
The point P, shown in Figure 1, is the minimum turning point on C. (b) Show that the x coordinate of P is a solution of x x = 4 - (3)
104. 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)
105. 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)
106. Oct 2020 · AS Paper 2 (Mechanics) · Q3(a) · 4 marks
3. A particle P moves along a straight line such that at time t seconds, t 0, after leaving the point O on the line, the velocity, v m s–1, of P is modelled as v = (7 – 2t)(t + 2) (a) Find the value of t at the instant when P stops accelerating. (4)
107. Oct 2020 · Paper 3 (Mechanics) · Q3(ii) · 6 marks
(ii) At time t seconds, where t 0 , a particle Q moves so that its position vector r metres, relative to a fixed origin O, is given by r = (t2 - t) i + 3t j Find the value of t at the instant when the speed of Q is 5 m s-1 (6)
108. June 2019 · Paper 1 · Q3(b) · 1 marks
y (b) Hence deduce the range of values for x for which d x < 0 d (1)
109. June 2019 · AS Paper 1 · Q5(a) · 3 marks
5. A curve has equation y = 3x2 + 24 x + 2 x < 0 (a) Find, in simplest form, d y d x (3)
110. June 2019 · AS Paper 1 · Q5(b) · 2 marks
(b) Hence find the exact range of values of x for which the curve is increasing. (2)
111. 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)
112. June 2019 · Paper 1 · Q12(c) · 3 marks
The function H(t) is used to model the height, in metres, of a ball above the ground t seconds after it has been kicked. Using this model, find (c) the maximum height of the ball above the ground between the first and second bounce. (3)
113. June 2019 · AS Paper 1 · Q13 · 7 marks
13. y R k x Figure 3 Figure 3 shows a sketch of part of the curve with equation y = 2x3 – 17x2 + 40x The curve has a minimum turning point at x = k. The region R, shown shaded in Figure 3, is bounded by the curve, the x-axis and the line with equation x = k. Show that the area of R is 256 (Solutions based entirely on graphical or numerical methods are not acceptable.) (7)
114. 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)
115. June 2019 · Paper 1 · Q14(b) · 2 marks
(b) (i) Use the small angle approximation for sin 2y to find an equation linking x and y for points close to the origin. (ii) Explain the relationship between the answers to (a) and (b)(i). (2)
116. June 2019 · Paper 2 · Q2(a) · 3 marks
2. The speed of a small jet aircraft was measured every 5 seconds, starting from the time it turned onto a runway, until the time when it left the ground. The results are given in the table below with the time in seconds and the speed in m sí1. Time (s) Speed (m sí1) Using all of this information, (a) estimate the length of runway used by the jet to take off. (3)
117. June 2019 · Paper 2 · Q2(b) · 1 marks
Given that the jet accelerated smoothly in these 25 seconds, (b) explain whether your answer to part (a) is an underestimate or an overestimate of the length of runway used by the jet to take off. (1)
118. 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)
119. June 2019 · Paper 2 · Q13(a) · 4 marks
13. r m h m r m Figure 9 [A sphere of radius r has volume 4 3 ʌU࣠3 and surface area 4ʌU࣠2] A manufacturer produces a storage tank. The tank is modelled in the shape of a hollow circular cylinder closed at one end with a hemispherical shell at the other end as shown in Figure 9. The walls of the tank are assumed to have negligible thickness. The cylinder has radius r metres and height h metres and the hemisphere has radius r metres. The volume of the tank is 6 m3. (a) Show that, according to the model, the surface area of the tank, in m2, is given by r + 5 3 ʌU࣠2 (4)
120. June 2019 · Paper 2 · Q13(b) · 4 marks
The manufacturer needs to minimise the surface area of the tank. (b) Use calculus to find the radius of the tank for which the surface area is a minimum. (4)
121. June 2019 · Paper 2 · Q13(c) · 2 marks
(c) Calculate the minimum surface area of the tank, giving your answer to the nearest integer. (2)
122. June 2019 · AS Paper 2 (Mechanics) · Q3(a) · 5 marks
3. A particle, P, moves along a straight line such that at time t seconds, t . 0, the velocity of P, v m s–1, is modelled as v = 12 + 4t – t2 Find (a) the magnitude of the acceleration of P when P is at instantaneous rest, (5)
123. June 2019 · Paper 3 (Mechanics) · Q1(a) · 3 marks
1. [In this question position vectors are given relative to afixed origin O| At time ¢ seconds, where ft > 0, a particle, P, moves so that its velocity vms! is given by 3 v = 6f1 -— 5Pj When f= 0, the position vector ofP is (—20i + 20j)m. (a) Find the acceleration of P when ¢ = 4 (3)
124. June 2018 · Paper 1 · Q2(a) · 3 marks
2. A curve C has equation y = x2 - 2x - 24 x, x > 0 d y (a) Find (i) d x d y (ii) d x (3)
125. June 2018 · Paper 1 · Q2(b) · 2 marks
(b) Verify that C has a stationary point when x = 4 (2)
126. June 2018 · Paper 1 · Q2(c) · 2 marks
(c) Determine the nature of this stationary point, giving a reason for your answer. (2)
127. June 2018 · AS Paper 1 · Q8(a) · 6 marks
8. A lorry is driven between London and Newcastle. In a simple model, the cost of the journey £C when the lorry is driven at a steady speed of v kilometres per hour is C = 1500 + 2 v + 60 v (a) Find, according to this model, (i) the value of v that minimises the cost of the journey, (ii) the minimum cost of the journey. (Solutions based entirely on graphical or numerical methods are not acceptable.) (6)
128. June 2018 · AS Paper 1 · Q8(b) · 2 marks
(b) Prove by using d C v that the cost is minimised at the speed found in (a)(i). d (2)
129. June 2018 · AS Paper 1 · Q8(c) · 1 marks
(c) State one limitation of this model. (1)
130. June 2018 · Paper 1 · Q9(c) · 1 marks
(c) Explain briefly how to find the coordinates of the point that is furthest north of the origin O. (You do not need to carry out this calculation). (1)
131. June 2018 · AS Paper 1 · Q10 · 4 marks
10. Prove, from first principles, that the derivative of x3 is 3x2 (4)
132. June 2018 · AS Paper 1 · Q15 · 10 marks
15. y Not to scale C P l R x Figure 4 Figure 4 shows a sketch of part of the curve C with equation y = 32 2x + 3x í 8, = = = = x < 0 The point P (4, 6) lies on C. The line l is the normal to C at the point P. The region R, shown shaded in Figure 4, is bounded by the line l, the curve C, the line with equation x = 2 and the x-axis. Show that the area of R is 46 (Solutions based entirely on graphical or numerical methods are not acceptable.) (10)
133. June 2018 · Paper 2 · Q5(a) · 3 marks
5. The equation 2x3 + x2 - 1 = 0 has exactly one real root. (a) Show that, for this equation, the Newton-Raphson formula can be written x + x + x + = n n n x + x n n (3)
134. June 2018 · Paper 2 · Q5(c) · 1 marks
(c) Explain why, for this question, the Newton-Raphson method cannot be used with x1 = 0 (1)
135. June 2018 · AS Paper 2 · Q8(a) · 5 marks
8. A particle, P, moves along the x-axis. At time t seconds, t . 0, the displacement, x metres, of P from the origin O, is given by x = t ( t - t + ) (a) Find the times when P is instantaneously at rest. (5)
136. 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)
137. 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)
138. 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)
139. 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)
140. 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 · AS Paper 1 · Q6
Mark scheme June 2025 Paper 1 Q6(b)
June 2025 · Paper 1 · Q7
Mark scheme June 2025 Paper 1 Q7(c)
June 2025 · Paper 1 · Q9
Mark scheme June 2025 Paper 1 Q9(b)
June 2025 · AS Paper 1 · Q11
Mark scheme June 2025 Paper 1 Q11(e)
June 2025 · Paper 1 · Q11
Mark scheme June 2025 Paper 1 Q11(b)
June 2025 · AS Paper 1 · Q12
Mark scheme June 2025 Paper 1 Q12(b)
June 2025 · AS Paper 1 · Q13
Mark scheme June 2025 Paper 1 Q13(d)(i)
June 2025 · Paper 1 · Q15
Mark scheme June 2025 Paper 1 Q15(c)
June 2025 · Paper 2 · Q8
Mark scheme June 2025 Paper 2 Q8(c)
June 2025 · Paper 2 · Q11
Mark scheme June 2025 Paper 2 Q11(d)
June 2025 · Paper 2 · Q15
Mark scheme June 2025 Paper 2 Q15(a)
June 2025 · AS Paper 2 (Mechanics) · Q4
Mark scheme June 2025 Paper 2 (Mechanics) Q4(c)
June 2025 · Paper 3 (Mechanics) · Q4
Mark scheme June 2025 Paper 3 (Mechanics) Q4(d)
June 2024 · Paper 1 · Q3
Mark scheme June 2024 Paper 1 Q3(b)
June 2024 · Paper 1 · Q4
Mark scheme June 2024 Paper 1 Q4
June 2024 · Paper 1 · Q5
Mark scheme June 2024 Paper 1 Q5(b)
June 2024 · AS Paper 1 · Q8
Mark scheme June 2024 Paper 1 Q8(a)
June 2024 · Paper 1 · Q10
Mark scheme June 2024 Paper 1 Q10(b)
June 2024 · AS Paper 1 · Q12
Mark scheme June 2024 Paper 1 Q12(d)
June 2024 · Paper 2 · Q1
Mark scheme June 2024 Paper 2 Q1(b)
June 2024 · Paper 2 · Q6
Mark scheme June 2024 Paper 2 Q6(a)
June 2024 · Paper 2 · Q9
Mark scheme June 2024 Paper 2 Q9(a)
June 2024 · Paper 2 · Q15
Mark scheme June 2024 Paper 2 Q15(b)
June 2024 · AS Paper 2 (Mechanics) · Q2
Mark scheme June 2024 Paper 2 (Mechanics) Q2(a)
June 2024 · Paper 3 (Mechanics) · Q4
Mark scheme June 2024 Paper 3 (Mechanics) Q4(c)
June 2024 · Paper 3 (Mechanics) · Q5
Mark scheme June 2024 Paper 3 (Mechanics) Q5(c)
June 2023 · AS Paper 1 · Q1
Mark scheme June 2023 Paper 1 Q1(b)
June 2023 · Paper 1 · Q4
Mark scheme June 2023 Paper 1 Q4(b)
June 2023 · AS Paper 1 · Q11
Mark scheme June 2023 Paper 1 Q11(b)
June 2023 · Paper 1 · Q12
Mark scheme June 2023 Paper 1 Q12
June 2023 · Paper 1 · Q15
Mark scheme June 2023 Paper 1 Q15(b)
June 2023 · AS Paper 1 · Q16
Mark scheme June 2023 Paper 1 Q16
June 2023 · Paper 2 · Q1
Mark scheme June 2023 Paper 2 Q1(b)
June 2023 · Paper 2 · Q4
Mark scheme June 2023 Paper 2 Q4(b)
June 2023 · Paper 2 · Q5
Mark scheme June 2023 Paper 2 Q5(b)
June 2023 · Paper 2 · Q7
Mark scheme June 2023 Paper 2 Q7(b)
June 2023 · Paper 2 · Q9
Mark scheme June 2023 Paper 2 Q9(b)
June 2023 · AS Paper 2 (Mechanics) · Q3
Mark scheme June 2023 Paper 2 (Mechanics) Q3(b)
June 2023 · Paper 3 (Mechanics) · Q3
Mark scheme June 2023 Paper 3 (Mechanics) Q3(c)
June 2022 · Paper 1 · Q6
Mark scheme June 2022 Paper 1 Q6(a)
June 2022 · AS Paper 1 · Q7
Mark scheme June 2022 Paper 1 Q7(c)
June 2022 · Paper 1 · Q8
Mark scheme June 2022 Paper 1 Q8(b)
June 2022 · AS Paper 1 · Q8
Mark scheme June 2022 Paper 1 Q8(c)
June 2022 · AS Paper 1 · Q10
Mark scheme June 2022 Paper 1 Q10(a)
June 2022 · AS Paper 1 · Q12
Mark scheme June 2022 Paper 1 Q12(d)
June 2022 · Paper 1 · Q15
Mark scheme June 2022 Paper 1 Q15(c)
June 2022 · Paper 1 · Q16
Mark scheme June 2022 Paper 1 Q16(a)
June 2022 · Paper 2 · Q4
Mark scheme June 2022 Paper 2 Q4
June 2022 · Paper 2 · Q6
Mark scheme June 2022 Paper 2 Q6(c)
June 2022 · Paper 2 · Q12
Mark scheme June 2022 Paper 2 Q12(b)
June 2022 · AS Paper 2 (Mechanics) · Q3
Mark scheme June 2022 Paper 2 (Mechanics) Q3(a)
June 2022 · Paper 3 (Mechanics) · Q1
Mark scheme June 2022 Paper 3 (Mechanics) Q1(b)
Oct 2021 · Paper 1 · Q4
Mark scheme Oct 2021 Paper 1 Q4(a)
Oct 2021 · AS Paper 1 · Q5
Mark scheme Oct 2021 Paper 1 Q5(c)
Oct 2021 · Paper 1 · Q8
Mark scheme Oct 2021 Paper 1 Q8(b)
Oct 2021 · Paper 1 · Q12
Mark scheme Oct 2021 Paper 1 Q12(a)
Oct 2021 · Paper 1 · Q14
Mark scheme Oct 2021 Paper 1 Q14
Oct 2021 · AS Paper 1 · Q14
Mark scheme Oct 2021 Paper 1 Q14(b)
Oct 2021 · AS Paper 1 · Q16
Mark scheme Oct 2021 Paper 1 Q16(b)
Oct 2021 · Paper 2 · Q5
Mark scheme Oct 2021 Paper 2 Q5(b)
Oct 2021 · Paper 2 · Q7
Mark scheme Oct 2021 Paper 2 Q7(a)
Oct 2021 · Paper 2 · Q8
Mark scheme Oct 2021 Paper 2 Q8(b)
Oct 2021 · Paper 2 · Q13
Mark scheme Oct 2021 Paper 2 Q13(a)
Oct 2021 · AS Paper 2 (Mechanics) · Q2
Mark scheme Oct 2021 Paper 2 (Mechanics) Q2(a)
Oct 2021 · Paper 3 (Mechanics) · Q5
Mark scheme Oct 2021 Paper 3 (Mechanics) Q5(a)
Oct 2020 · AS Paper 1 · Q1
Mark scheme Oct 2020 Paper 1 Q1
Oct 2020 · Paper 1 · Q9
Mark scheme Oct 2020 Paper 1 Q9(c)
Oct 2020 · AS Paper 1 · Q14
Mark scheme Oct 2020 Paper 1 Q14(b)
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(b)
Oct 2020 · Paper 2 · Q12
Mark scheme Oct 2020 Paper 2 Q12(a)(i)
Oct 2020 · Paper 2 · Q13
Mark scheme Oct 2020 Paper 2 Q13(b)
Oct 2020 · AS Paper 2 (Mechanics) · Q3
Mark scheme Oct 2020 Paper 2 (Mechanics) Q3(a)
Oct 2020 · Paper 3 (Mechanics) · Q3
Mark scheme Oct 2020 Paper 3 (Mechanics) Q3(ii)
June 2019 · Paper 1 · Q3
Mark scheme June 2019 Paper 1 Q3(b)
June 2019 · AS Paper 1 · Q5
Mark scheme June 2019 Paper 1 Q5(b)
June 2019 · Paper 1 · Q12
Mark scheme June 2019 Paper 1 Q12(c)
June 2019 · AS Paper 1 · Q13
Mark scheme June 2019 Paper 1 Q13
June 2019 · Paper 1 · Q14
Mark scheme June 2019 Paper 1 Q14(b)
June 2019 · Paper 2 · Q2
Mark scheme June 2019 Paper 2 Q2(b)
June 2019 · Paper 2 · Q11
Mark scheme June 2019 Paper 2 Q11(a)
June 2019 · Paper 2 · Q13
Mark scheme June 2019 Paper 2 Q13(c)
June 2019 · AS Paper 2 (Mechanics) · Q3
Mark scheme June 2019 Paper 2 (Mechanics) Q3(a)
June 2019 · Paper 3 (Mechanics) · Q1
Mark scheme June 2019 Paper 3 (Mechanics) Q1(a)
June 2018 · Paper 1 · Q2
Mark scheme June 2018 Paper 1 Q2(c)
June 2018 · AS Paper 1 · Q8
Mark scheme June 2018 Paper 1 Q8(c)
June 2018 · Paper 1 · Q9
Mark scheme June 2018 Paper 1 Q9(c)
June 2018 · AS Paper 1 · Q10
Mark scheme June 2018 Paper 1 Q10
June 2018 · AS Paper 1 · Q15
Mark scheme June 2018 Paper 1 Q15
June 2018 · Paper 2 · Q5
Mark scheme June 2018 Paper 2 Q5(c)
June 2018 · AS Paper 2 · Q8
Mark scheme June 2018 Paper 2 Q8(a)
June 2018 · Paper 2 · Q9
Mark scheme June 2018 Paper 2 Q9
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(c)