plumiThe quadratic function: Edexcel International GCSE Further Maths questions
Edexcel International GCSE Further Maths past-paper questions on 2 The quadratic function, 2019–2025, with mark schemes and examiner’s reports where available.
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All 42 questions on The quadratic function, as a list
- June 2025 (R), Paper 1, Q8 (15 marks): f x x 2 x 2 4 9 The curve C has equation y f x The line l with equation y 8 x 9 0 is a tangent to C The line k is perpendicular to l and is also the tangent to
- June 2025, Paper 1, Q5 (8 marks): (a) Show that ( 3 3 3 ) 3 ( ) (2) The quadratic equation 2 x 2 x has roots α and β 6 7 0 Without solving the equation 2 (b) form a quadratic equation, with inte
- November 2024, Paper 1, Q10 (18 marks): x x x 2 f ( ) 3 4 9 Given that f ( ) x can be expressed in the form A B x ( C )2 where A, B and C are positive constants (a) find the value of A, the value of B
- June 2024 (R), Paper 1, Q10 (12 marks): The quadratic equation 2 x kx has roots α and β such that 2 4 0 k < 0 and α > β 7 17 Given that α β 2 2 4 (a) show that k 7 (8) (b) Hence form a quadratic equat
- June 2024 (R), Paper 2, Q1 (4 marks): The quadratic equation kx k x 2 2 6 16 0 has equal roots. Find the two possible values of k (4)
- June 2024, Paper 1, Q2 (5 marks): x x x 2 f( ) 2 4 9 2 Given that f( ) x can be written in the form A x B C , where A, B and C are integers, (a) find the value of A, the value of B and the value
- June 2024, Paper 2, Q2 (8 marks): The quadratic equation 3 has roots x x 2 5 1 0 and Without solving the equation, form a quadratic equation with integer coefficients, that has roots and 2 2 (8)
- November 2023, Paper 1, Q2 (7 marks): 1 2 g x x 2 x 2 3 2 2 (a) Express g(x) in the form p x q r where p, q and r are rational numbers to be found. (3) (b) Find (i) the minimum value of g(x) (ii) th
- November 2023, Paper 2, Q1 (5 marks): The equation where k is a constant, has real unequal roots. kx 2 8 x 3 k 0 Find the set of values of k giving your answer in an exact simplified form. (5)
- November 2023, Paper 2, Q10 (10 marks): The roots of a quadratic equation are and where 5 115 3 3 and 2 8 (a) Show that 4 (3) (b) Form a quadratic equation with integer coefficients, that has roots 2
- June 2023 (R), Paper 2, Q1 (4 marks): f x 2 x 2 k 8 x k Show that for all values of k, the equation f x 0 has distinct real roots. (4)
- June 2023 (R), Paper 2, Q6 (10 marks): f (x) = 2 x x p 2 5 The equation f (x) = 0 has roots α and β 215 Given that α 3 β 3 8 (a) find the value of p (5) Without solving the equation f (x) = 0 (b) for
- June 2023, Paper 1, Q11 (14 marks): f (x) = 10 + 6x – x 2 Given that f (x) can be written in the form A(x + B) 2 + C where A, B and C are constants, (a) find the value of A, the value of B and the
- June 2023, Paper 2, Q11 (12 marks): The roots of a quadratic equation E are α and β where α > β > 0 Given that α – β = 2 6 and α2 + β2 = 30 (a) show that (i) αβ = 3 (4) (ii) α + β = 6 (2) (b) With
- January 2023 (R), Paper 2, Q8 (14 marks): The quadratic equation 3x2 – kx – 1 = 0 , where k is a positive integer, has roots α and β (a) Show that α2 + β2 = k 2 + 6 9 (3) Given that α4 + β4 = 466 81 (b)
- January 2023, Paper 1, Q5 (8 marks): Given that k is a non‑zero constant curve C has equation kx2 – xy + (k + 1)x = 1 straight line l has equation y = k 2 x + 1 The point A is the only point that l
- January 2023, Paper 2, Q3 (13 marks): f(x) = 8x2 + 10x – 3 Given that f(x) can be written in the form A(x + B)2 + C where A, B and C are constants, (a) find the value of A, the value of B and the va
- January 2023, Paper 2, Q8 (11 marks): The quadratic equation x2 – 4k 2 x + 2k 4 – 1 = 0 where k is a positive constant, has roots α and β Given that α 2 + β 2 = 66 and that α 3 + β 3 = p 2 where p i
- June 2022 (R), Paper 1, Q8 (10 marks): (a) Using the binomial expansion, or otherwise, find the complete expansion of (x + y)3 (1) The quadratic equation 2x2 + 3x + 4 = 0 has roots α and β (b) Withou
- June 2022, Paper 2, Q1 (4 marks): Find the set of values of k for which the equation 2kx2 + 5kx + 5k – 3 = 0 where k ≠ 0 has real roots. (4)
- June 2022, Paper 2, Q5 (9 marks): The roots of the quadratic equation 2x2 + (6 + 2p) x + 2p = 0 are α and β (a) Write down an expression in terms of p for (i) α + β (ii) αβ (2) (b) Show that (α
- January 2022 (R), Paper 2, Q4 (10 marks): The quadratic equation 2x2 + 4x + 3 = 0 has roots α and β (a) Without solving the equation, show that α2 + β2 = 1 (4) (b) Without solving the equation, find the
- January 2022, Paper 1, Q2 (7 marks): f(x) = 2x 2 – 12x + 5 Given that f(x) can be written in the form f(x) = a(x + b) 2 + c where a, b and c are integers, (a) find the value of a, the value of b an
- January 2022, Paper 2, Q7 (13 marks): f(x) = 2x 2 + px + q where p and q are integers g(x) = 14x 2 + 37x + 14 The equation f(x) = 0 has roots α and β The equation g(x) = 0 has roots α and β β α Give
- November 2021, Paper 1, Q1 (7 marks): The roots of the equation 4x2 – 3x – 8 = 0 are α and β Without solving this equation, form a quadratic equation, with integer coefficients, which has roots 1 α
- November 2021, Paper 1, Q2 (5 marks): f(x) = 2x2 + ( p – 1)x – 2p where p is a constant. Find the set of values of p for which the equation f(x) = 0 has two distinct real roots. (5)
- June 2021, Paper 1, Q1 (6 marks): The quadratic equation 3(k + 2)x2 + (k +5)x + k = 0 has real roots. Find the set of possible values of k. (6)
- June 2021, Paper 1, Q6 (10 marks): (a) Show that (α – β)2 = (α + β)2 – 4αβ (3) The quadratic equation x2 – 7kx + k2 = 0, where k is a positive constant, has roots α and β where α > β (b) Show tha
- June 2021, Paper 2, Q2 (6 marks): f(x) = 2 + 4 5 x – 1 2 25 x2 Given that f(x) can be expressed in the form A – B(x + C)2 where A, B and C are constants, (a) find the value of A, the value of B
- November 2020 (R), Paper 1, Q8 (7 marks): f (x) = 3x2 − x + 4 g(x) = x2 − px + q The roots of the quadratic equation f (x) = 0 are α and β 1 1 The roots of the quadratic equation g(x) = 0 are α + and β
- November 2020 (R), Paper 2, Q7 (12 marks): (a) Find the set of values of k for which the equation kx2 – 4x + 2k = 7 has real roots (4) Given that the roots of the equation kx2 – 4x + 2k = 7 are α and β,
- November 2020, Paper 1, Q7 (13 marks): f(x) = x2 − 9x + 14 Given that f(x) can be written in the form (x + a)2 + b, where a and b are constants, (a) find the value of a and the value of b. (2) (b) He
- November 2020, Paper 2, Q6 (11 marks): f(x) = 4x2 – 3x – 5 The equation f(x) = 0 has roots α and β Without solving the equation f(x) = 0 (a) form an equation, with integer coefficients, that has root
- January 2020 (R), Paper 1, Q2 (5 marks): f(x) = 7 + 4x – x2 (a) Write f(x) in the form a – b(x + c)2 where a, b and c are integers to be found. (3) (b) Hence, or otherwise, find (i) the maximum value o
- January 2020 (R), Paper 2, Q5 (12 marks): (a) Show that (α + β )3 − 3αβ (α + β ) = α3 + β 3 (2) The roots of the equation 2x2 + 3x + 6 = 0 are α and β Without solving the equation, (b) find the value of
- January 2020, Paper 2, Q5 (8 marks): Given that α and β are such that α + β = 7 2 and αβ = 2 (a) form a quadratic equation with integer coefficients that has roots α and β, (2) (b) form a quadratic
- June 2019 (R), Paper 1, Q10 (13 marks): f (x x í x2 x ∈ 10 Given that f (x) can be written in the form D(x + E)2 + F where D, E and F are integers, (a) find the value of D, the value of E and the valu
- June 2019 (R), Paper 2, Q1 (7 marks): f (x) = (x í [x2 p í x + q] 1 *LYHQ WKDW I í (a) find the value of q. (2) (b) Find the range of values of p for which the cubic equation f (x) = 0 has only one
- June 2019 (R), Paper 2, Q11 (12 marks): The quadratic equation x2 í px + q = 0 where p 0 , has roots Į and ȕ. Given that 2Įȕ = 3 and that 4(Į2 + ȕ2) = k2 í k í ZKHUH k 3 (a) (i) write down the value
- June 2019, Paper 1, Q5 (5 marks): f(x)= 3x-9x+5 5 Given that f(x) can be written in the form a(x-b) +c, where a, b and c are constants, find (a) the value of a, the value of b and the value of c
- June 2019, Paper 1, Q8 (10 marks): (a) Solve 5p' - 9p + 4 = 0 (2) (b) Hence solve 52x+1 - 9(5*) + 4 = 0 Give your answers to 3 significant figures where appropriate. (4) The curve with equation y
- June 2019, Paper 2, Q10 (15 marks): The roots of the equation x2 + 3x – 5 = 0 are Į and ȕ. (a) Without solving the equation, find (i) the value of Į2 + ȕ2 (ii) the value of Į4 + ȕ4 (5) Given that
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