Roots of polynomials: OCR A‑Level Further Maths questions

OCR A-Level Further Maths past-paper questions on 4.1 Roots of polynomials, 2018–2025, with mark schemes and examiner’s reports where available. Add MadAsMaths questions to your practice, or browse them separately by topic and difficulty.

All 18 questions on Roots of polynomials, as a list

June 2025, Pure Core 1, Q4 (9 marks): 2 4 The equation x 5 4 x 10 0 has roots a, b and c. - + = (a) Write down the values of a + , ab and abc. [2] + + + b c bc ca )2 2 2 2 2 2 2 (b) By expanding ( ,
June 2025, AS Pure Core, Q3 (6 marks): + 3 The roots of the equation x x 2 3 5 0 are denoted by a and b. + = (a) Write down the value of a [2] + and the value of ab. b (b) Using the answers to part (
June 2024, Pure Core 1, Q4 (3 marks): In this question you must show detailed reasoning. 3 2 The equation x 2 3 x 6 x 3 0 has roots a, b and c. + + - = 2 2 2 Determine a cubic equation with integer
June 2024, AS Pure Core, Q7 (6 marks): In this question you must show detailed reasoning. 3 2 The roots of the equation x 2 3 x 3 x 5 0 are a, b and c. - - + = )2 )2 By considering ( and ( , determin
June 2023, Pure Core 1, Q2 (6 marks): In this question you must show detailed reasoning. 4 3 2 The equation z 4 z 9 z 10 z 6 0 has roots a, b, c and d. + + + + = (a) Show that a quartic equation who
June 2023, AS Pure Core, Q1 (5 marks): 3 1 The roots of the equation x 4 2 x 3 x 2 0 are a, b, c and d. By using a suitable - - + = substitution, find a quartic equation whose roots are a + , 2 b + ,
June 2023, AS Pure Core, Q5 (4 marks): In this question you must show detailed reasoning. 2 - The roots of the equation x x = are a and b. 5 3 12 0 + By considering the symmetric functions of the roo
June 2022, Pure Core 1, Q9 (5 marks): The cube roots of unity are represented on the Argand diagram below by the points A, B and C. Im(z) B L O M A Re(z) N C The points L, M and N are the midpoints
June 2022, Pure Core 2, Q3 (6 marks): In this question you must show detailed reasoning. 3 2 The roots of the equation x 4 6 x 3 x 9 0 are a, b and c. + - + = [6] Find a cubic equation with integer
June 2022, AS Pure Core, Q3 (6 marks): In this question you must show detailed reasoning. 3 2 The roots of the equation x 5 3 x 2 x 9 0 are a, b and c. - - + = Find a cubic equation with integer coef
November 2021, Pure Core 1, Q3 (8 marks): 2 3 A function f(z) is defined on all complex numbers z by ( )z f z 3 z kz - where k is a real 5 = - + 2 2 2 constant. The roots of the equation ( )z f 0 are a,
November 2021, AS Pure Core, Q2 (4 marks): 2 2 The equation x 2 3 x 2 x 5 0 has roots a, b and c. + - + = Use a substitution to find a cubic equation with integer coefficients whose roots are a + , 1 b +
November 2020, Pure Core 1, Q9 (9 marks): 2 9 You are given that the cubic equation 2 x px qx 3 0 , where p and q are real numbers, has a + + - = complex root 1 i 2 . a = + (a) Write down a second compl
November 2020, Pure Core 2, Q2 (6 marks): In this question you must show detailed reasoning. 3 2 The roots of the equation x 3 2 x 5 x 4 0 are a, b and c. - - - = 2 2 2 (a) Find a cubic equation with in
November 2020, AS Pure Core, Q5 (7 marks): In this question you must show detailed reasoning. 3 2 The cubic equation x 5 3 x 4 x 7 0 has roots a, b and c. + - + = Find a cubic equation with integer coeff
June 2019, Pure Core 1, Q1 (4 marks): In this question you must show detailed reasoning. 2 - The quadratic equation x 2 x 5 0 has roots a and . + = b (a) Write down the values of a + and ab . [1] b
June 2019, AS Pure Core, Q5 (9 marks): In this question you must show detailed reasoning. 3 2 You are given that a, b and c are the roots of the equation x 5 2 x 3 x 1 0 . - + + = 2 2 2 2 2 2 (a) Fin
June 2018, AS Pure Core, Q2 (3 marks): In this question you must show detailed reasoning. 3 2 The cubic equation x x x 2 3 5 4 0 has roots a, b and c. By making an appropriate substitution, or + - +

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