Further algebra and functions: AQA A‑Level Further Maths questions

Every AQA A-Level Further Maths exam question on D Further algebra and functions, 2019–2025, with its mark scheme and examiner’s report.

All 92 questions on Further algebra and functions, as a list

June 2025, Paper 1, Q3 (1 mark): Which one of the following expressions cannot be evaluated by l’Hôpital’s rule? Circle your answer. [1 mark] ( ) e 2 0 0 0 0 tanh 1 lim lim lim cos sin 2 ln lim
June 2025, Paper 1, Q7 (4 marks): The cubic equation + − + = 3 2 5 4 3 0 z z z has roots α, β and γ Find an equation, with integer coefficients, that has roots − 2 1 α , − 2 1 β and − 2 1 γ [4 m
June 2025, Paper 1, Q8 (3 marks): The curve 1 C has equation + = 2 2 1 4 25 x y The curve 1 C is translated by the vector    1 3 to give the curve 2 C Find the coordinates of the points wh
June 2025, Paper 1, Q11 (7 marks): The function f is defined by ( ) ( ) f e 1 1 x x x = ∈ +  11 (a) Show that ( ) ( ) e e f e 2 3 1 x x x '' x − + = + [3 marks] 11 (b) Hence, find the Maclaurin
June 2025, Paper 1, Q13 (7 marks): The function f is defined by f ( ) = + + + 3 2 4 92 z z rz z s where r and s are real numbers. One of the roots of the equation f ( ) = 0 z is i −+ 4 3 13 (a) F
June 2025, Paper 2, Q8 (5 marks): The function f is defined by f ( ) = + − 3 2 12 x x x x ( ) x∈ Figure 1 shows the graph of ( ) f y x = Figure 1 x O y 8 (a) The graph of ( ) f y x = is transfo
June 2025, Paper 2, Q10 (13 marks): The function f is defined by ( ) f( ) − = + + 2 2 2 4 6 9 x x x x 10 (a) Write down the equations of the asymptotes to the graph of f ( ) y x = [2 marks] 10 (b)
June 2025, AS Paper 1, Q3 (1 mark): Find the equations of the asymptotes of the curve with equation y = (x – 1)(x + 2) (x + 1)(x – 2) Tick () one box. [1 mark] x = 1, x = –2, y = 1 x = 1, x = –2,
June 2025, AS Paper 1, Q4 (1 mark): Find the first two non-zero terms of the Maclaurin series expansion of cos (2x) Tick () one box. [1 mark] 1 – x2 1 – 2x2 2 – x2 2 – 2x2
June 2025, AS Paper 1, Q5 (2 marks): The quartic equation 3x4 + x2 – 5x – 11 = 0 has roots α, β, γ and δ 5 (a) Write down the value of αβγδ [1 mark] 5 (b) Write down the value of αβγ + αβδ + αγδ +
June 2025, AS Paper 1, Q8 (7 marks): (a) Show that, for all positive integers r, 1 r2 – 1 (r + 1)2 = 2r + 1 r2 (r + 1)2 [1 mark] 8 (b) Hence, using the method of differences, show that n ∑ r =1 2r
June 2025, AS Paper 1, Q9 (4 marks): Solve the inequality 2x + 1 3x – 9 ≥ x – 1 [4 marks]
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, Q1 (1 mark): The roots of the equation 20x3 – 16x2 – 4x + 7 = 0 are α, β and γ Find the value of αβ + βγ + γα Circle your answer. [1 mark] _ – 4 5 – 1 5 1 5 4 5
June 2024, Paper 1, Q8 (4 marks): The ellipse E has equation _ x2 + y2 9 = 1 The line with equation y = mx + 4 is a tangent to E Without using differentiation show that m = ±√ 7 [4 marks]
June 2024, Paper 2, Q5 (3 marks): The first four terms of the series S can be written as S = (1 × 2) + (2 × 3) + (3 × 4) + (4 × 5) + ... 5 (a) Write an expression, using ∑ notation, for the sum
June 2024, Paper 2, Q6 (3 marks): The cubic equation x3 + 5x2 – 4x + 2 = 0 has roots α, β and γ Find a cubic equation, with integer coefficients, whose roots are 3α, 3β and 3γ [3 marks]
June 2024, Paper 2, Q13 (8 marks): (a) Use the method of differences to show that n ∑ r =2 1 (r – 1)r(r + 1) = 1 4 – 1 2n + 1 2(n + 1) [5 marks] 13 (b) Find the smallest integer n such that n ∑ r
June 2024, Paper 2, Q15 (7 marks): The diagram shows the line y = 5 – x O y x 15 (a) On the diagram above, sketch the graph of y = │x2 – 4x│, including all parts of the graph where it intersects
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, Q18 (4 marks): In this question you may use results from the formulae booklet without proof. Use the binomial series for (1 + x)n and the Maclaurin’s series for sin x to find
June 2024, AS Paper 1, Q9 (7 marks): (a) Show that, for all positive integers r, r + 1 r + 2 – r r + 1 = 1 (r + 1)(r + 2) [1 mark] ��
June 2024, AS Paper 1, Q10 (6 marks): The curve C has equation y = 2x – 10 3x – 5 Figure 1 shows the curve C with its asymptotes. Figure 1 x 5 y O 2 10 (a) Write down the equations of the asymptotes
June 2024, AS Paper 1, Q13 (5 marks): The cubic equation x3 – x – 7 = 0 has roots α, β and γ The cubic equation p(x) = 0 has roots α – 1, β – 1 and γ – 1 The coefficient of x3 in p(x) is 1 13 (a) De
June 2024, AS Paper 1, Q15 (7 marks): (a) Use Maclaurin’s series expansion for In(1 + x) to show that the first three terms of the Maclaurin’s series expansion of In(1 + 3x) are 3x – 9 2 x2 + 9x3 [1
June 2024, AS Paper 1, Q16 (6 marks): The curve C has the polar equation r = 2 √cos2 θ + 4 sin2 θ –π < θ ≤ π 16 (a) Show that the Cartesian equation of C can be written as x2 a2 + y2 b2 = 1 where a
June 2023, Paper 1, Q5 (6 marks): The function f is defined by f (r) = 2r(r − 2) (r 2 Z) 5 (a) Show that f (r + 1) − f (r) = r2r [2 marks] 5 (b) Use the method of differences to show that X n r=
June 2023, Paper 1, Q7 (5 marks): The function f is defined by f (x) = sinx + 1 2 (0 × x × 2 ) Find the set of values of x for which f (x) × 1 2 Give your answer in set notation. [5 marks]
June 2023, Paper 1, Q11 (7 marks): The function f is defined by f (x) = 4x3 − 8x2 − 51x − 45 (x 2 R) 11 (a) (i) Fully factorise f (x) [2 marks] 11 (a) (ii) Hence, solve the inequality f (x) < 0 [
June 2023, Paper 1, Q13 (5 marks): Use l’Hoˆ pital’s rule to prove that lim x ! x sin 2x cos x 2 0 B @ 1 C A = −4 [5 marks]
June 2023, Paper 1, Q14 (10 marks): The curve C has polar equation r = 4 5 + 3 cosy ( − < y × ) 14 (a) Show that r takes values in the range 1 k × r × k , where k is an integer. [2 marks] 14 (b) F
June 2023, Paper 2, Q5 (5 marks): Josh and Zoe are solving the following mathematics problem: The curve C1 has equation x2 16 − y2 9 = 1 The matrix M = 0 1 1 0 " # maps C1 onto C2 Find the equat
June 2023, Paper 2, Q7 (3 marks): Show that X n+1 r=11 r3 = 1 4 (n2 + an + b)(n2 + an + c) where a, b and c are integers to be found. [3 marks]
June 2023, Paper 2, Q13 (11 marks): The quadratic equation z2 − 5z + 8 = 0 has roots a and b 13 (a) Write down the value of a + b and the value of ab [2 marks] 13 (b) Without finding the value of
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, Q4 (1 mark): The roots of the equation 5x3 + 2x2 – 3x + p = 0 are α, β and γ Given that p is a constant, state the value of αβ + βγ + γα Circle your answer. [1 mark] – 3 5 –
June 2023, AS Paper 1, Q6 (6 marks): (a) Find and simplify the first five terms in the Maclaurin series for e2x [2 marks] 6 (b) Hence, or otherwise, write down the first five terms in the Maclaurin
June 2023, AS Paper 1, Q7 (7 marks): (a) Show that, for all integers r, 1 2r – 1 – 1 2r + 1 = 2 (2r – 1)(2r + 1) [1 mark] 7 (b) Hence, using the method of differences, show that n  r =1 1 (2r – 1)
June 2023, AS Paper 1, Q10 (9 marks): The curve C has equation y = 3x2 + mx + p x2 + px + m where m and p are integers. The vertical asymptotes of C are x = – 4 and x = –1 The curve C is shown in th
June 2023, AS Paper 1, Q11 (8 marks): A point has Cartesian coordinates (x, y) and polar coordinates (r, θ) where r ≥ 0 and – π < θ ≤ π 11 (a) Express r in terms of x and y [1 mark] 11 (b) Express x
June 2023, AS Paper 1, Q13 (10 marks): (a) Prove by induction that, for all integers n ≥ 1, n  r =1 r2 = 1 6 n(n + 1)(2n + 1) [4 marks] 13 (c) Use the formula in part (a) to write down a factorised
June 2023, AS Paper 1, Q14 (4 marks): The inequality (x2 – 5x – 24)(x2 + 7x + a) < 0 has the solution set {x : – 9 < x < – 3} ∩ {x : 2 < x < b} Find the values of integers a and b [4 marks]
June 2022, Paper 1, Q5 (6 marks): It is given that z = − 3 2 + i 11 p 2 is a root of the equation z4 − 3z3 − 5z2 + kz + 40 = 0 where k is a real number. 5 (a) Find the other three roots. [5 mark
June 2022, Paper 2, Q3 (1 mark): The roots of the equation x2 − px − 6 = 0 are a and b Find a2 + b2 in terms of p Circle your answer. [1 mark] p2 − 6 p2 + 6 p2 − 12 p2 + 12
June 2022, Paper 2, Q7 (8 marks): The function f is defined by f (x) = ax − 5 2x + b x 2 R, x 6= 9 2 where a and b are integers. The graph of y = f (x) has asymptotes x = 9 2 and y = 3 7 (a) Fin
June 2022, Paper 2, Q8 (10 marks): (a) The function f is defined as f (x) = sec x 8 (a) (i) Show that f (4)(0) = 5 [4 marks] 8 (a) (ii) Hence find the first three non-zero terms of the Maclaurin
June 2022, Paper 2, Q10 (7 marks): The curve C1 has equation x2 25 − y2 4 = 1 The curve C2 has equation x2 − 25y2 − 6x − 200y − 416 = 0 10 (a) Find a sequence of transformations that maps the gra
June 2022, AS Paper 1, Q2 (1 mark): The quadratic equation x2 + px + q = 0 has roots a and b Which of the following is equal to ab ? Circle your answer. [1 mark] p −p q −q 1 0 0 2 3
June 2022, AS Paper 1, Q9 (5 marks): (a) Show that, for r > 0, ln (r + 2) − ln r = ln 1 + 2 r [1 mark] 9 (b) Hence, using the method of differences, show that X n r=1 ln 1 + 2 r = ln 1 2 (n + a)(n
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
June 2022, AS Paper 1, Q13 (10 marks): A curve C1 has equation y = 2x + 7 3x + 5 13 (a) Write down the equations of the asymptotes of curve C1 [2 marks] 13 (b) On the axes below, sketch the graph of
June 2022, AS Paper 1, Q14 (15 marks): The function f is defined by f (x) = x2 − 3 x2 + px + 7 x 2 R where p is a constant. The graph of y = f (x) has only one asymptote. 14 (a) Write down the equati
November 2021, Paper 1, Q1 (1 mark): Find X 20 r=1 (r2 − 2r) Circle your answer. [1 mark] 2450 2660 5320 43680
November 2021, Paper 1, Q7 (7 marks): The diagram below shows the graph of y = f (x) ( − 4 ≥ x ≥ 4) The graph meets the x-axis at x = 1 and x = 3 The graph meets the y-axis at y = 2 x 4 – 4 3 1 2 y
November 2021, Paper 1, Q9 (4 marks): Use l’Hoˆ pital’s rule to show that lim x ! 1(xe−x) = 0 Fully justify your answer. [4 marks]
November 2021, Paper 1, Q13 (3 marks): The transformation S is represented by the matrix 3 0 0 1 The transformation T is a translation by the vector 0 −5 Kamla transforms the graphs of various functi
November 2021, Paper 2, Q4 (7 marks): (a) Show that (r + 1)2 − r2 = 2r + 1 [1 mark] 4 (b) Use the method of differences to show that X n r=1 (2r + 1) = n2 + 2n [3 marks] 4 (c) Verify that using the
November 2021, Paper 2, Q5 (5 marks): The equation z3 + 2z2 − 5z − 3 = 0 has roots a, b and g Find a cubic equation with roots 1 2 a − 1, 1 2 b − 1 and 1 2 g − 1 [5 marks]
November 2021, Paper 2, Q6 (8 marks): The ellipse E1 has equation x2 + y2 4 = 1 E1 is translated by the vector 3 0 to give the ellipse E2 6 (a) Write down the equation of E2 [1 mark] 6 (b) The ellip
November 2021, AS Paper 1, Q7 (3 marks): Show that the Maclaurin series for ln (e + 2ex) is 1 + 2x − 2x2 + ax3 − ::: where a is to be determined. [3 marks]
November 2021, AS Paper 1, Q8 (6 marks): Stephen is correctly told that (1 + i) and −1 are two roots of the polynomial equation z3 − 2iz2 + pz + q = 0 where p and q are complex numbers. 8 (a) Stephen s
November 2021, AS Paper 1, Q9 (7 marks): (a) Use the standard formulae for X n r=1 r and X n r=1 r2 to show that X n r=1 r(r + 3) = an(n + 1)(n + b) where a and b are constants to be determined. [4 mar
November 2021, AS Paper 1, Q11 (4 marks): (a) Show that, for all positive integers r, 1 (r − 1)! − 1 r! = r − 1 r! [1 mark] 11 (b) Hence, using the method of differences, show that X n r=1 r − 1 r! = a
November 2021, AS Paper 1, Q14 (9 marks): Curve C1 has equation x2 16 + y2 4 = 1 14 (a) Curve C2 is a reflection of C1 in the line y = x Write down an equation of C2 [1 mark] 14 (b) Curve C3 is a circle
November 2021, AS Paper 1, Q16 (8 marks): Curve C has equation y = ax x + b where a and b are constants. The equations of the asymptotes to C are x = −2 and y = 3 x y y = ax x + b O 16 (a) Write down th
November 2020, Paper 1, Q3 (1 mark): The quadratic equation ax2 + bx + c = 0 (a, b, c 2 R) has real roots a and b. One of the four statements below is incorrect. Which statement is incorrect? Tick
November 2020, Paper 1, Q5 (9 marks): H1 is the locus of points such that the distance from the point (5, 0) is twice the distance from the line x = 2 5 (a) Show that the equation of H1 can be writt
November 2020, Paper 1, Q8 (6 marks): The three roots of the equation 4x3 − 12x2 − 13x + k = 0 where k is a constant, form an arithmetic sequence. Find the roots of the equation. [6 marks]
November 2020, Paper 1, Q9 (13 marks): The function f is defined by f (x) = x(x + 3) x + 4 (x 2 R, x 6= −4) 9 (a) Find the interval (a, b) in which f (x) does not take any values. Fully justify your
November 2020, Paper 1, Q14 (6 marks): (a) Given that sinh (A + B) = sinh A cosh B + coshA sinhB express sinh (m + 1)x and sinh (m − 1)x in terms of sinhmx, coshmx, sinhx and coshx [1 mark] 14 (b) He
November 2020, Paper 2, Q5 (5 marks): Solve the inequality 2x + 3 x × 1 × x + 5 [5 marks]
November 2020, Paper 2, Q6 (5 marks): Find the sum of all the integers from 1 to 999 inclusive that are not square or cube numbers. [5 marks]
November 2020, Paper 2, Q11 (8 marks): (a) Starting from the series given in the formulae booklet, show that the general term of the Maclaurin series for sin x x × cos x is (×1)r+1 2r (2r + 1)!x2r [4
November 2020, AS Paper 1, Q3 (1 mark): Given (x − 1)(x − 2)(x − a) < 0 and a > 2 Find the set of possible values of x. Tick (3) one box. [1 mark] fx : x < 1g ¨ fx : 2 < x < ag fx : 1 < x < 2g ¨ fx :
November 2020, AS Paper 1, Q5 (4 marks): (a) Show that r2(r + 1)2 − (r − 1)2r2 = pr3 where p is an integer to be found. [1 mark] 5 (b) Hence use the method of differences to show that X n r=1 r3 = 1 4
November 2020, AS Paper 1, Q9 (8 marks): The quadratic equation 2x2 + px + 3 = 0 has two roots, a and b, where a > b. 9 (a) (i) Write down the value of ab. [1 mark] 9 (a) (ii) Express a + b in terms of
November 2020, AS Paper 1, Q10 (8 marks): (a) Show that the equation y = 3x − 5 2x + 4 can be written in the form (x + a)(y + b) = c where a, b and c are integers to be found. [3 marks] 10 (b) Write dow
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, Q14 (7 marks): (a) Given x + 7 x + 1 × x + 1 show that (x + a)(x + b) x + c × 0 where a, b, and c are integers to be found. [4 marks] 14 (b) Briefly explain why this statement
June 2019, Paper 1, Q2 (1 mark): The first two non-zero terms of the Maclaurin series expansion of f (x) are x and − 1 2x3 Which one of the following could be f (x)? Circle your answer. [1 mark
June 2019, Paper 1, Q13 (14 marks): The equation z3 + kz2 + 9 = 0 has roots a, b and g. 13 (a) (i) Show that a2 + b2 + g2 = k 2 [3 marks] 13 (a) (ii) Show that a2b2 + b2g2 + g2a2 = −18k [4 marks]
June 2019, Paper 2, Q2 (1 mark): Which of the straight lines given below is an asymptote to the curve y = ax2 x − 1 where a is a non-zero constant? Circle your answer. [1 mark] y = ax + a y = a
June 2019, Paper 2, Q3 (1 mark): The set A is defined by A = fx : −p2 < x < 0g ¨ fx : 0 < x < p2g Which of the inequalities given below has A as its solution? Circle your answer. [1 mark] jx2 −
June 2019, Paper 2, Q4 (3 marks): The positive integer k is such that X k r=1 (3r − k) = 90 Find the value of k. [3 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, Q14 (12 marks): Let Sn = X n r=1 1 (r + 1)(r + 3) where n × 1 14 (a) Use the method of differences to show that Sn = 5n2 + an 12(n + b)(n + c) where a, b and c are integers. [6
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
June 2019, AS Paper 1, Q6 (5 marks): (a) On the axes provided, sketch the graph of x = cosh (y + b) where b is a positive constant. [4 marks] x y O 6 (b) Determine the minimum distance between the
June 2019, AS Paper 1, Q7 (5 marks): (a) Show that 1 r − 1 − 1 r + 1 × A r2 − 1 where A is a constant to be found. [1 mark] 7 (b) Hence use the method of differences to show that X n r=2 1 r2 − 1 ×
June 2019, AS Paper 1, Q10 (6 marks): (a) Using the definition of coshx and the Maclaurin series expansion of ex, find the first three non-zero terms in the Maclaurin series expansion of cosh x. [3
June 2019, AS Paper 1, Q11 (8 marks): (a) Curve C has equation y = x2 + px − q x2 − r where p, q and r are positive constants. Write down the equations of its asymptotes. [2 marks] 11 (b) Find the s
June 2019, AS Paper 1, Q14 (7 marks): The graph of y = x3 − 3x is shown below. x y y = x3 – 3x O The two stationary points have x-coordinates of −1 and 1 The cubic equation x3 − 3x + p = 0 where p i

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