plumiFurther calculus: Edexcel A‑Level Further Maths questions
Every Edexcel A-Level Further Maths exam question on CP.5 Further calculus, 2018–2025, with its mark scheme and examiner’s report.
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All 27 questions on Further calculus, as a list
- June 2025, Core Pure 1, Q7 (8 marks): (a) Express 2 10 9 22 2 3 3 2 2 x x x x x in partial fractions. (4) (b) Hence, show that 2 10 9 22 2 3 2 27 4 6 3 3 2 2 0 1 x x x x x x d ln ϖ (4)
- June 2025, Core Pure 1, Q10 (12 marks): (a) Given that, for n ∈ z z n n n 1 2cos θ z z n n n 1 2isin θ show that 8 4 4 2 3 4 sin cos cos θ θ θ (5) O y R x Figure 1 Figure 2 Figure 1 shows the central
- June 2025, Core Pure 2, Q9 (7 marks): Given that y x = arcsin3 −1 3 1 3 x (a) determine d d y x in terms of x (2) The curve C has equation y x cos arcsin3 −1 3 1 3 x (b) Determine d d y x gi
- June 2025, AS Core Pure, Q7 (8 marks): x O y C –12 24 m Figure 1 Figure 2 In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not accept
- June 2024, Core Pure 2, Q3 (5 marks): (a) Explain why 1 9 16 2 4 3 x x d is an improper integral. (1) (b) Show that 2 4 3 1 d 9 16 x kπ x where k is a constant to be determined. (4)
- June 2024, Core Pure 2, Q9 (10 marks): A 4 cm 4.5 cm D E F B C D O C x y 7 cm Figure 1 Figure 2 Figure 1 shows the central vertical cross-section ABCDEFA of a vase together with measurements that hav
- June 2024, AS Core Pure, Q8 (11 marks): x E D y B A O R C B A Figure 1 Figure 2 O C Figure 1 shows the central vertical cross-section, OABCDEO, of the design for a solid glass ornament. Figure 2 shows
- June 2023, Core Pure 1, Q2 (6 marks): (a) Write x 2 + 4x – 5 in the form (x + p) 2 + q where p and q are integers. (1) (b) Hence use a standard integral from the formula book to find 1 4 5 2x x x d
- June 2023, Core Pure 2, Q7 (8 marks): In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. 1.257 –1.545 O x y Figure 2 J
- June 2023, AS Core Pure, Q5 (8 marks): In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Figure 1 O C R y x Figure 2 A large pi
- June 2022, Core Pure 1, Q6 (7 marks): (a) Express as partial fractions 2 3 6 1 4 2 2 x x x x + + + ( ) + ( ) (3) (b) Hence, show that ( )( ) ( ) 2 2 2 0 2 3 6 d ln 2 1 4 x x x a bπ x x + + = + + + ∫
- June 2022, Core Pure 1, Q8 (12 marks): (a) Given zn + 1 zn = 2 cos nθ n ∈ show that 32 cos6 θ ≡ cos 6θ + 6 cos 4θ + 15 cos 2θ + 10 (5) y x O x = –4 x = 4 R Figure 1 Figure 2 Figure 1 shows a solid
- June 2022, Core Pure 1, Q9 (6 marks): (i) (a) Explain why cosh x x 0 ∞∫ d is an improper integral. (1) (b) Show that cosh x x 0 ∞∫ d is divergent. (3) (ii) 4 sinh x = p cosh x where p is a real cons
- June 2022, Core Pure 2, Q5 (6 marks): (a) Given that y = arcsin x –1 x 1 show that d d y x x = − 1 1 2 (3) (b) f(x) = arcsin (e x) x 0 Prove that f(x) has no stationary points. (3)
- June 2022, AS Core Pure, Q8 (15 marks): 16 cm 8 cm (−8, a) (8, a) y x O (0, a) top base Figure 1 Figure 2 Figure 1 shows a sketch of a 16 cm tall vase which has a flat circular base with diameter 8 cm
- November 2021, Core Pure 1, Q5 (7 marks): (i) Evaluate the improper integral 2 1 1 2 e d ∞ − ∫ x x (3) (ii) The air temperature, θ °C, on a particular day in London is modelled by the equation θ = 8 – 5
- November 2021, Core Pure 1, Q9 (11 marks): (a) Use a hyperbolic substitution and calculus to show that x x x x x x k 2 2 2 1 1 2 1 − = − + + ∫ d arcosh where k is an arbitrary constant. (6) y O
- November 2021, Core Pure 2, Q5 (8 marks): The curve C has equation y = arccos 1 2 x −2 x 2 (a) Show that C has no stationary points. (3) The normal to C, at the point where x = 1 , crosses
- November 2021, Core Pure 2, Q7 (9 marks): Solutions based entirely on graphical or numerical methods are not acceptable. O y x R y = arsinh x y = β Figure 1 Figure 1 shows a sketch of part of the curve
- November 2020, Core Pure 1, Q2 (7 marks): (a) Explain why ( ) 1 1 d 2 5 x x x + ∫ is an improper integral. (1) (b) Prove that ( ) 1 1 d ln 2 5 x a b x x = + ∫ where a and b are rational numbers to b
- November 2020, Core Pure 2, Q5 (10 marks): (a) y = tan−1 x Assuming the derivative of tan x , prove that d d y x x = + 1 1 2 (3) f ( x ) = x tan−1 4x (b) Show that f d ( ) tan tan x x Ax x Bx C x k ∫ = +
- November 2020, Core Pure 2, Q7 (11 marks): y x O A C1 B C2 Figure 1 A student wants to make plastic chess pieces using a 3D printer. Figure 1 shows the central vertical cross-section of the student’s des
- November 2020, AS Core Pure, Q3 (5 marks): y O x R Figure 1 Figure 1 shows a circle with radius r and centre at the origin. The region R, shown shaded in Figure 1, is bounded by the x-axis and the part o
- June 2019, Core Pure 1, Q2 (7 marks): Show that ( )( ) 2 0 8 12 d ln 2 3 1 x x k x x − = + + ∫ where kLVDUDWLRQDOQXPEHUWREHIRXQG (7)
- June 2019, Core Pure 2, Q3 (6 marks): f ( ) x x = + 1 4 9 2 (a) Using a substitution, that should be stated clearly, show that f (x)dx = A sinh–1 (Bx) + c where c is an arbitrary constant and A an
- June 2019, Core Pure 2, Q8 (11 marks): A C B D 2.36 m 2 m y x O 1 B D Figure 1 Figure 2 Figure 1 shows the central vertical cross section ABCD of a paddling pool that has a circular horizontal cross
- June 2018, AS Core Pure, Q9 (11 marks): 2 cm 10 cm 4 cm 14 cm 8 cm A H B G C F D E y G F E O x 1 4 Figure 1 Figure 2 A mathematics student is modelling the profile of a glass bottle of water. Figure 1
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