plumiFurther calculus: AQA A‑Level Further Maths questions
Every AQA A-Level Further Maths exam question on E Further calculus, 2019–2025, with its mark scheme and examiner’s report.
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All 37 questions on Further calculus, as a list
- June 2025, Paper 1, Q16 (7 marks): The picture shows a solar cooker. Part of the solar cooker is a parabolic dish. The shape of the internal surface of the dish is formed by rotating the part of
- June 2025, Paper 2, Q3 (1 mark): Find ( ) d d − − − 1 1 sin cos 2 x x x Circle your answer. [1 mark] − − 2 3 1 x − − 2 1 1 x − 2 1 1 x − 2 3 1 x
- June 2025, Paper 2, Q4 (1 mark): The function f is defined by f ( ) = − 2 16 x x ( ) x∈ On which of the following intervals is the mean value of f the greatest? Tick () one box. [1 mark] 0 ≤
- June 2025, Paper 2, Q15 (6 marks): Prove that ( )( ) d 5 1 13 ln 1 2 3 12 x x x ∞ = + + ∫ Show the limiting process clearly. [6 marks]
- June 2025, Paper 2, Q16 (5 marks): The diagram shows a design for a table top. x y O The table top is modelled as being bounded by the lines . = 0 5 x and . = 1 5 x , and the curves defined by .
- June 2025, AS Paper 1, Q13 (5 marks): The curve C1 is given by the equation xy = m where m is a positive constant. The region R1 is enclosed by curve C1, the x-axis and the lines x = 1 and x = 4 O 1
- June 2024, Paper 1, Q3 (1 mark): The function f is defined by _ f (x) = x2 (x ∈ ℝ) Find the mean value of f (x) between x = 0 and x = 2 Circle your answer. [1 mark] 2 3 4 3 8 3 16 3
- June 2024, Paper 1, Q4 (1 mark): Which one of the following statements is correct? Tick () one box. [1 mark] lim x 0(x2 ln x) = 0 lim x 0(x2 ln x) = 1 lim x 0(x2 ln x) = 2 lim x 0(x2 ln x) is
- June 2024, Paper 1, Q11 (5 marks): (a) Find d dx (x2tan–1 x) [1 mark] 11 (b) Hence find ∫2x tan–1 x dx [4 marks]
- June 2024, Paper 1, Q15 (5 marks): A curve is defined parametrically by the equations x = 3 2t3 + 5 y = t 9 2 (t ≥ 0) Show that the arc length of the curve from t = 0 to t = 2 is equal to 26 unit
- June 2024, Paper 2, Q16 (9 marks): The function f is defined by f(x) = ax + 5 x + b where a and b are constants. The graph of y = f(x) has asymptotes x = –2 and y = 3 16 (a) Write down the value
- June 2024, Paper 2, Q20 (9 marks): The integral In is defined by In = ∫0 π 4 cosn x dx (n ≥ 0) 20 (a) Show that In = ( n – 1) n In–2 + 1 n(2 n 2) (n ≥ 2) [6 marks] 20 (b) Use the result from part
- June 2024, AS Paper 1, Q2 (1 mark): The function f is defined by f(x) = 2x + 3 0 ≤ x ≤ 5 The region R is enclosed by y = f(x), x = 5, the x-axis and the y-axis. The region R is rotated through 2π
- June 2024, AS Paper 1, Q7 (3 marks): The function f is defined by f(x) = 1 √ x 4 ≤ x ≤ 7 Find the mean value of f over the interval 4 ≤ x ≤ 7 Give your answer in exact form. [3 marks] �������������
- June 2023, Paper 1, Q16 (11 marks): (a) Show that ( 4 0:5 1 t ln t dt = a(ln 2)2 where a is a rational number to be found. [4 marks] 16 (b) A curve C is defined parametrically for t > 0 by x = 2t
- June 2023, Paper 2, Q2 (1 mark): Which one of the expressions below is not equal to zero? Circle your answer. [1 mark] lim x ! 1(x2e−x) lim x ! 0(x5 ln x) lim x ! 1 ex x5 lim x ! 0(x3ex)
- June 2023, Paper 2, Q14 (10 marks): The function f is defined by f (x) = 1 4x2 + 16x + 19 (x 2 R) 14 (a) Show, without using calculus, that the graph of y = f (x) has a stationary point at −2, 1 3
- June 2023, AS Paper 1, Q5 (4 marks): The function f is defined by f (x) = 3x2 1 ≤ x ≤ 5 5 (a) Find the mean value of f [2 marks] 5 (b) The function g is defined by g (x) = f (x) + c 1 ≤ x ≤ 5 The m
- June 2022, Paper 2, Q2 (1 mark): Find the mean value of the function f (x) = 10x4 between x = 0 and x = a Circle your answer. [1 mark] 10a3 40a3 2a4 4a5
- June 2022, Paper 2, Q12 (11 marks): The shaded region shown in the diagram below is bounded by the x-axis, the curve y = f (x) , and the lines x = a and x = b O a b y x y = f(x) The shaded region
- June 2022, AS Paper 1, Q10 (6 marks): The diagram below shows an ellipse E The coordinate axes are the lines of symmetry of E O x y 2 – 2 – 3 3 10 (a) Write down an equation of E [2 marks] 10 (b) Th
- November 2021, Paper 1, Q10 (6 marks): Evaluate the improper integral ( 8 0 ln x dx showing the limiting process. [6 marks]
- November 2021, Paper 1, Q14 (12 marks): The hyperbola H has equation y2 − x2 = 16 The circle C has equation x2 + y2 = 32 The diagram below shows part of the graph of H and part of the graph of C. x y
- November 2021, Paper 2, Q7 (7 marks): y t = 0 x t = 2 The diagram shows a curve known as an astroid. The curve has parametric equations x = 4 cos3 t y = 4 sin3 t (0 × t < 2p) The section of the curv
- November 2021, Paper 2, Q12 (12 marks): The integral Sn is defined by Sn = (a 0 xn sinhx dx (n ≤ 0) 12 (a) Show that for n ≤ 2 Sn = n(n − 1)Sn−2 + an cosh a − nan−1 sinha [7 marks] 12 (b) Hence show t
- November 2021, AS Paper 1, Q2 (1 mark): Given that f (x) = 3x − 1 find the mean value of f (x) over the interval 4 × x × 8 Circle your answer. [1 mark] 6 11 17 23
- November 2020, Paper 1, Q1 (1 mark): Which of the integrals below is not an improper integral? Circle your answer. [1 mark] ( 1 0 e−x dx ( 2 0 1 1 − x2 dx ( 1 0 x p dx ( 1 0 1x p dx
- November 2020, Paper 1, Q10 (10 marks): (a) Find the general solution of the differential equation dy dx + 2y x = x + 3 x(x − 1)(x2 + 3) (x > 1) [8 marks] 10 (b) Find the particular solution for which
- November 2020, Paper 2, Q3 (1 mark): Find the gradient of the tangent to the curve y = sin×1 x at the point where x = 1 5 Circle your answer. [1 mark] 5 6 p 12 2 6 p 5 4 3 p 25 25 24 for the next q
- November 2020, Paper 2, Q7 (5 marks): The diagram shows part of the graph of y = cos×1 x 0.2 O 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 – 0.2 – 0.2 0.2 0.4 0.6 0.8 1.0 1.2 x y The finite region enclosed by t
- November 2020, AS Paper 1, Q12 (2 marks): The mean value of the function f over the interval 1 × x × 5 is m. The graph of y = g(x) is a reflection in the x-axis of y = f (x) . The graph of y = h(x) is a
- November 2020, AS Paper 1, Q15 (4 marks): A segment of the line y = kx is rotated about the x-axis to generate a cone with vertex O. The distance of O from the centre of the base of the cone is h. The r
- June 2019, Paper 1, Q3 (1 mark): The function f (x) = x2 − 1 Find the mean value of f (x) from x = −0:5 to x = 1:7 Give your answer to three significant figures. Circle your answer. [1 mark] −0
- June 2019, Paper 2, Q5 (4 marks): A curve has equation y = coshx Show that the arc length of the curve from x = a to x = b , where 0 < a < b , is equal to sinh b − sinha [4 marks]
- June 2019, Paper 2, Q8 (9 marks): A parabola P1 has equation y2 = 4ax where a > 0 P1 is translated by the vector b 0 , where b > 0 , to give the parabola P2 8 (a) The line y = mx is a tangent to
- June 2019, Paper 2, Q13 (10 marks): (a) Explain why (1 3 x2e−2x dx is an improper integral. [1 mark] 13 (b) Evaluate (1 3 x2e−2x dx Show the limiting process. [9 marks]
- June 2019, AS Paper 1, Q5 (8 marks): A hyperbola H has the equation x2 a2 − y2 4a2 = 1 where a is a positive constant. 5 (a) Write down the equations of the asymptotes of H. [1 mark] 5 (b) Sketch t
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