Tangents, normals and stationary points: Edexcel A‑Level Maths questions

Edexcel A-Level Maths past-paper questions on 7.3 Tangents, normals and stationary points, 2010–2025, with mark schemes and examiner’s reports where available. Add MadAsMaths questions to your practice, or browse them separately by topic and difficulty.

All 38 questions on Tangents, normals and stationary points, as a list

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, Q9 (9 marks): O T t v 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
June 2025, Pure 1, Q15 (9 marks): O D C B A Figure 4 Figure 4 shows the plan view for the design of a stage. The shape of this design consists of a sector of a circle AOB joined to a rectangle O
June 2025, Pure 2, Q8 (9 marks): O C y x Figure 2 f x x x x ax 4 3 2 1 3 8 17 3 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
June 2025, AS Pure, Q11 (13 marks): In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. r cm l cm Figure 4 Figure 4 s
June 2025, AS Pure, Q12 (5 marks): y x O C 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 D
June 2024, Pure 1, Q5 (6 marks): The function f is defined by f(x) = 2 2 3 4 x x x (a) Show that f′(x) = ax bx c x 2 2 2 4 where a, b and c are constants to be found. (3) (b) Hence, using a
June 2024, Pure 1, Q10 (9 marks): A l1 R l2 O x y Figure 3 In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Figu
June 2024, Pure 2, Q15 (12 marks): The curve C has equation ( ) x y x y 3 2 3 3 2 (a) Find an expression for d d y x in terms of x and y. (5) The point P(1, 0) lies on C. (b) Show that the normal
June 2024, AS Pure, Q8 (10 marks): A B C R O y x l Figure 3 In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Figu
June 2024, AS Pure, Q12 (13 marks): x y A B Figure 5 Figure 5 shows the plan view of the design for a swimming pool. The pool is modelled as a quarter of a circle joined to two equal sized rectang
June 2023, Pure 1, Q4 (5 marks): 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 =
June 2023, Pure 2, Q5 (5 marks): 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 d d y x x x x k 2 9 5 3 2 where k is a c
June 2023, Pure 2, Q7 (7 marks): 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
June 2023, AS Pure, Q1 (6 marks): A curve has equation (a) Find d d y x writing your answer in simplest form. (2) (b) Hence find the range of values of x for which y is decreasing. (4)
June 2023, AS Pure, Q16 (6 marks): A curve has equation y x x = f ( ), 0 Given that • f ( ) x x a x b 4 , where a and b are constants • the curve has a stationary point at 4 3 , • the curve meet
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, Q8 (8 marks): v t T O 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 traffi
June 2022, Pure 1, Q15 (10 marks): A 0.8 rad h r r F E D C B Figure 5 A company makes toys for children. Figure 5 shows the design for a solid toy that looks like a piece of cheese. The toy is mo
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.
June 2022, AS Pure, Q10 (10 marks): P C O R y x l Figure 2 In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Figure 2 shows
June 2022, AS Pure, Q12 (12 marks): A company makes drinks containers out of metal. The containers are modelled as closed cylinders with base radius r cm and height h cm and the capacity of each c
November 2021, Pure 2, Q5 (7 marks): The curve C has equation y = 5x4 – 24x3 + 42x2 – 32x + 11 x ∈  (a) Find (i) d d y x (ii) d d 2 2 y x (3) (b) (i) Verify that C has a stationary point at x = 1
November 2021, Pure 2, Q7 (9 marks): In this question you should show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. x C O l R P(5, –13) y Figur
November 2021, Pure 2, Q8 (9 marks): The curve C has equation px3 + qxy + 3y2 = 26 where p and q are constants. (a) Show that d d y x apx bqy qx cy 2 where a, b and c are integers to be found. (4)
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, 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, Q7 (10 marks): y P O x Figure 1 Figure 1 shows a sketch of the curve C with equation y = 4 2 2x x x + − 4 ln x x > 0 (a) Show that d d y x = 12 16 4 2x x x x x + − (4) The poi
November 2020, AS Pure, Q1 (5 marks): 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
November 2020, AS Pure, Q14 (9 marks): 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
June 2019, Pure 1, Q3 (5 marks): y x x x x = + + ≠− 5 10 1 1 2 2 ( ) (a) Show that d d y x A x n = + ( )1 where A and n are constants to be found. (4) (b) Hence deduce the range of values for x
June 2019, Pure 1, Q12 (10 marks): f(x) = 10e±௘x 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) y O
June 2019, Pure 2, Q13 (10 marks): r m r m h 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
June 2019, AS Pure, Q5 (5 marks): A curve has equation y = 3x2 + 24 x + 2 x 0 (a) Find, in simplest form, d d y x (3) (b) Hence find the exact range of values of x for which the curve is increas
June 2019, AS Pure, Q13 (7 marks): y x O k R 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,
June 2018, Pure 1, Q2 (7 marks): A curve C has equation y = x2 − 2x − 24 x, x > 0 (a) Find (i) d d y x (ii) d d 2 2 y x (3) (b) Verify that C has a stationary point when x = 4 (2) (c) Determine
June 2018, Pure 1, Q9 (10 marks): y Q O P East North x Figure 4 Figure 4 shows a sketch of the curve with equation x2 − 2xy + 3y2 = 50 (a) Show that d d y x y x y x = − − 3 (4) The curve is used
June 2018, Pure 2, Q11 (7 marks): 1 11 6 3 1 2 3 1 2 2 + − − − ≡ + − + − x x x x A B x C x ( )( ) ( ) ( ) (a) Find the values of the constants A, B and C. (4) f (x) = 1 11 6 3 1 2 2 + − − − x x

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