plumiAlgebra and functions: Edexcel A‑Level Maths questions
Every Edexcel A-Level Maths exam question on P.2 Algebra and functions, 2019–2025, with its mark scheme and examiner’s report.
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All 72 questions on Algebra and functions, as a list
- June 2025, Pure 1, Q1 (4 marks): The point P(6, –4) lies on the curve with equation y = f(x), x ∈ Find the point to which P is mapped when the curve with equation y = f(x) is transformed to t
- June 2025, Pure 1, Q4 (5 marks): Given that • f(x) = 2x 3 + 3x 2 – 16x + 16 • f(–4) = 0 (a) write f(x) in the form (x + a) Q(x) where a is a constant and Q(x) is a quadratic expression. (3) (b)
- June 2025, Pure 1, Q6 (7 marks): A scientist is monitoring the flight of sea birds after they leave their nests on a cliff. The height above the sea, h metres, of one of the birds is modelled b
- June 2025, Pure 1, Q7 (7 marks): O 2 –1 –75 5 C x y 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 turnin
- June 2025, Pure 1, Q12 (8 marks): O Q C l x y Figure 3 Figure 3 shows a sketch of the curve C with equation y = 15 2 3 3 x x x ( )( ) x ≠ – 3 2 x ≠ 3 and the straight line l with equation y = 2x
- June 2025, Pure 2, Q1 (3 marks): In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Factorise completely 2x 3 – 24x 2 + 40
- June 2025, Pure 2, Q9 (8 marks): In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. A new type of car is released
- June 2025, Pure 2, Q11 (11 marks): A company is trying to determine the most profitable selling price for a new toy. Given that • if the selling price of each toy is £30, the company expects to s
- June 2025, Pure 2, Q14 (8 marks): A B C D Not to scale Figure 4 In this question you must show detailed reasoning. Figure 4 shows a trapezium ABCD where AD is parallel to BC Given that • AB → =
- June 2025, Pure 2, Q15 (13 marks): R y x P Q O –1 1 Figure 5 In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Figure 5 sho
- June 2025, AS Pure, Q1 (6 marks): y x B (2, 7) A (–1, 0) O y = 3 Figure 1 Figure 1 shows a curve with equation y = f (x) The curve • passes through the point A (–1, 0) • has a maximum turning po
- June 2025, AS Pure, Q5 (6 marks): f (x) = x x x 5 2 12 4 − (a) Write f (x) in the form ax p + bx q where a, b, p and q are simplified constants. (3) (b) Hence find ∫f (x) dx giving your answer i
- June 2025, AS Pure, Q6 (7 marks): f (x) = ax 3 + bx 2 + 18x + 9 where a and b are constants. Given that (x + 3) is a factor of f (x) (a) show that –3a + b = 5 (2) Given also that f′ (2) = 14 (b)
- June 2025, AS Pure, Q7 (10 marks): P x O Figure 3 A company makes a particular type of chair. The annual profit made by the company is modelled by the equation P = –x 2 + 260x – 16 450 where P is
- June 2025, AS Pure, Q9 (3 marks): An equation x 2 – 3 px + 5 q + 4 = 0 where p and q are constants, has equal roots. Show that q = 1 20 (3 p + 4)(3 p – 4) (3)
- June 2024, Pure 1, Q1 (3 marks): g(x) = x x k x k 3 20 17 3 2 where k is a constant. Given that (x – 3) is a factor of g(x), find the value of k. (3)
- June 2024, Pure 1, Q6 (6 marks): O P x y Figure 1 Figure 1 shows a sketch of the graph with equation y x 3 2 5 The vertex of the graph is at the point P, shown in Figure 1. (a) Find the coordin
- June 2024, Pure 1, Q8 (11 marks): The functions f and g are defined by f(x) 2 4 3 5 9 , 2 9 2 x x x x x g(x) (a) Find fg(2) (2) (b) Find g –1 (3) (c) (i) Find gf(x), giving your answer as a simp
- June 2024, Pure 2, Q3 (4 marks): The point P(3, –2) lies on the curve with equation y = f(x), x ∈ Find the coordinates of the point to which P is mapped when the curve with equation y = f(x)
- June 2024, Pure 2, Q9 (7 marks): O B(20, 0.8) A(0, 2) H (m) 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. Th
- June 2024, Pure 2, Q12 (12 marks): (a) Express 1 25 V V in partial fractions. (2) The volume, V microlitres, of a plant cell t hours after the plant is watered is modelled by the differential equ
- June 2024, AS Pure, Q2 (8 marks): In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. f (x) = 2x 3 – 3ax 2 + bx + 8
- June 2024, AS Pure, Q5 (9 marks): The curve C1 has equation y = 6 x + 3 (a) (i) Sketch C1 stating the coordinates of any points where the curve cuts the coordinate axes. (ii) State the equations
- June 2023, Pure 1, Q2 (6 marks): In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. f(x) = 4x 3 + 5x 2 – 10x + 4a
- June 2023, Pure 1, Q7 (8 marks): The function f is defined by f(x) = 3 + x −2 x ∈ x > 2 (a) State the range of f (1) (b) Find f –1 (3) The function g is defined by g(x) = 15 3 x − x ∈ x ≠ 3
- June 2023, Pure 1, Q13 (7 marks): On a roller coaster ride, passengers travel in carriages around a track. On the ride, carriages complete multiple circuits of the track such that • the maximum
- June 2023, Pure 2, Q10 (7 marks): f ( ) ( )( ) x kx x x 3 18 4 2 where k is a positive constant (a) Express f (x) in partial fractions in terms of k. (3) (b) Hence find the exact value of k for
- June 2023, Pure 2, Q12 (10 marks): O 5 t N NA NB Figure 2 The number of subscribers to two different music streaming companies is being monitored. The number of subscribers, NA , in thousands, to
- June 2023, AS Pure, Q2 (4 marks): In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Using the substitution u x = or otherw
- June 2023, AS Pure, Q4 (5 marks): (a) Sketch the curve with equation y k x x 0 where k is a positive constant. (2) (b) Hence or otherwise, solve 16 2 x (3)
- June 2023, AS Pure, Q8 (5 marks): O R C l 6 (10, 80) 60 y x Figure 3 Figure 3 shows a sketch of a curve C and a straight line l. Given that • C has equation y x f where f (x) is a quadratic expr
- June 2023, AS Pure, Q15 (7 marks): In this question you must show detailed reasoning. Solutions relying on calculator technology are not acceptable. The curve C1 has equation The curve C2 has equ
- June 2022, Pure 1, Q1 (4 marks): The point P (−2, −5) lies on the curve with equation y = f (x), x ∈ Find the point to which P is mapped, when the curve with equation y = f (x) is transformed
- June 2022, Pure 1, Q2 (3 marks): f (x) = (x − 4)(x2 − 3x + k) − 42 where k is a constant Given that (x + 2) is a factor of f (x) , find the value of k. (3)
- June 2022, Pure 1, Q5 (6 marks): The height, h metres, of a tree, t years after being planted, is modelled by the equation h 2 = at + b 0 t < 25 where a and b are constants. Given that • the
- June 2022, Pure 1, Q6 (6 marks): (2, 8) C x 6 O y 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 or
- June 2022, Pure 1, Q11 (7 marks): y x C1 O C2 Figure 4 Figure 4 shows a sketch of part of the curve C1 with equation y = 2x3 + 10 x > 0 and part of the curve C2 with equation y = 42x − 15x2 − 7
- June 2022, Pure 2, Q1 (4 marks): In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. y = | 3 − 2x | y x O Figure 1
- June 2022, Pure 2, Q10 (8 marks): The function f is defined by f(x) = 8 5 2 3 x x + + x > −3 2 (a) Find f −1 3 2 (2) (b) Show that f(x) = A + B x 2 3 + where A and B are constants to b
- June 2022, Pure 2, Q14 (10 marks): (a) Express 3 2 1 1 ( )( ) x x − + in partial fractions. (3) When chemical A and chemical B are mixed, oxygen is produced. A scientist mixed these two chemicals
- June 2022, AS Pure, Q2 (7 marks): f(x) = 2x3 + 5x2 + 2x + 15 (a) Use the factor theorem to show that (x + 3) is a factor of f(x). (2) (b) Find the constants a, b and c such that f(x) = (x + 3)(a
- June 2022, AS Pure, Q7 (7 marks): (a) Factorise completely 9x – x3 (2) The curve C has equation y = 9x – x3 (b) Sketch C showing the coordinates of the points at which the curve cuts the x‑axis.
- November 2021, Pure 1, Q1 (3 marks): f (x) = ax3 + 10x2 − 3ax − 4 Given that (x − 1) is a factor of f (x), find the value of the constant a. You must make your method clear. (3)
- November 2021, Pure 1, Q2 (4 marks): Given that f (x) = x2 − 4x + 5 x ∈ (a) express f (x) in the form (x + a)2 + b where a and b are integers to be found. (2) The curve with equation y = f (x) •
- November 2021, Pure 1, Q9 (11 marks): f(x) = 50 38 9 5 2 1 2 2 2 x x x x + + + ( ) − ( ) x ≠ −2 5 x ≠ 1 2 Given that f(x) can be expressed in the form A x B x C x 5 2 5 2 1 2 2 + + + ( ) + − where A
- November 2021, Pure 1, Q12 (9 marks): H x O 3 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,
- November 2021, Pure 2, Q2 (5 marks): The functions f and g are defined by f (x) = 7 – 2x2 x Î g (x) = 3 5 1 x x − x Î x ≠1 5 (a) State the range of f (1) (b) Find gf (1.8) (2) (c) Find g–1(x) (
- November 2021, Pure 2, Q11 (10 marks): y x O Figure 4 Figure 4 shows a sketch of the graph with equation y = | 2x – 3k | where k is a positive constant. (a) Sketch the graph with equation y = f (x) w
- November 2021, AS Pure, Q1 (3 marks): In this question you should show all stages of your working. Solutions relying on calculator technology are not acceptable. Using algebra, solve the inequality
- November 2021, AS Pure, Q2 (3 marks): In this question you should show all stages of your working. Solutions relying on calculator technology are not acceptable. Given 9 3 81 1 2 x y − + = express y
- November 2021, AS Pure, Q6 (6 marks): In this question you should show all stages of your working. Solutions relying on calculator technology are not acceptable. (a) Using algebra, find all solution
- November 2021, AS Pure, Q14 (10 marks): A curve C has equation y = f(x) where f(x) = –3x 2 + 12x + 8 (a) Write f(x) in the form a(x + b) 2 + c where a, b and c are constants to be found. (3) The curve
- November 2021, AS Pure, Q16 (11 marks): 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
- November 2020, Pure 1, Q4 (5 marks): The function f is defined by f (x) = 3 7 2 x x − − x , x ≠ 2 (a) Find f −1(7) (2) (b) Show that ff (x) = ax b x + −3 where a and b are integers to be found.
- November 2020, Pure 1, Q7 (5 marks): y x O l C R 25 (−2, 13) Figure 1 Figure 1 shows a sketch of a curve C with equation y = f (x) and a straight line l. The curve C meets l at the points (−2, 13)
- November 2020, Pure 1, Q9 (9 marks): y x O C 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) (
- November 2020, Pure 2, Q5 (4 marks): The curve with equation y = 3 × 2 x meets the curve with equation y = 15 − 2 x+1 at the point P. Find, using algebra, the exact x coordinate of P. (4)
- November 2020, Pure 2, Q6 (7 marks): (a) Given that x x x Ax B C x x x 2 8 3 2 2 2 + − + ≡ + + + ∈ ≠− find the values of the constants A, B and C (3) (b) Hence, using algebraic integration, find
- November 2020, Pure 2, Q8 (6 marks): A curve C has equation y = f (x) Given that ● f ʹ (x) = 6x2 + ax − 23 where a is a constant ● the y intercept of C is −12 ● (x + 4) is a factor of f (x) find, i
- November 2020, Pure 2, Q11 (7 marks): y O P x Figure 2 Figure 2 shows a sketch of the graph with equation y = 2 | x + 4 | − 5 The vertex of the graph is at the point P, shown in Figure 2. (a) Find t
- November 2020, AS Pure, Q3 (6 marks): In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. (i) Solve the equation x 2 18 − = x wr
- November 2020, AS Pure, Q10 (10 marks): g(x) = 2x3 + x2 – 41x – 70 (a) Use the factor theorem to show that g(x) is divisible by (x – 5). (2) (b) Hence, showing all your working, write g(x) as a produc
- June 2019, Pure 1, Q1 (3 marks): f(x) = 3x3 + 2ax2 – 4x + 5a Given that (x + 3) is a factor of f(x), find the value of the constant a. (3)
- June 2019, Pure 1, Q5 (10 marks): f(x) = 2x2 + 4x + 9 x (a) Write f(x) in the form a(x + b)2 + c, where a, b and c are integers to be found. (3) (b) Sketch the curve with equation y = f(x) sho
- June 2019, Pure 1, Q13 (11 marks): The curve C with equation y p x x q x = − − + 3 2 3 ( )( ) x , x x ≠− ≠ 3 2 , where S and q are constants, passes through the point 3 1 2 , ⎛ ⎝⎜ ⎞ ⎠⎟ and has
- June 2019, Pure 2, Q1 (3 marks): Given 2 4 1 2 2 x y × = express y as a function of x. (3)
- June 2019, Pure 2, Q6 (10 marks): y O x Figure 4 Figure 4 shows a sketch of the graph of y J [࣠) , where g( ) ( ) x x x x x = − + − ⎧ ⎨ ⎩ 2 1 4 7 2 2 2 - (a) Find the value of gg(0). (2) (b)
- June 2019, Pure 2, Q7 (7 marks): A small factory makes bars of soap. On any day, the total cost to the factory, £y, of making x bars of soap is modelled to be the sum of two separate elements:
- June 2019, AS Pure, Q2 (8 marks): Find, using algebra, all real solutions to the equation (i) 16a2 = 2 a (4) (ii) b4 + 7b2 – 18 = 0 (4)
- June 2019, AS Pure, Q7 (8 marks): The curve C has equation y = k x 2 + 1 x \, x 0 where k is a constant. (a) Sketch C stating the equation of the horizontal asymptote. (3) The line l has equ
- June 2019, AS Pure, Q9 (6 marks): A company started mining tin in Riverdale on 1st January 2019. A model to find the total mass of tin that will be mined by the company in Riverdale is given by
- June 2019, AS Pure, Q11 (10 marks): f(x) = 2x3 – 13x2 + 8x + 48 (a) Prove that (x – 4) is a factor of f(x). (2) (b) Hence, using algebra, show that the equation f(x) = 0 has only two distinct root
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