Roots of polynomials: OCR MEI A‑Level Further Maths questions

OCR MEI 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 17 questions on Roots of polynomials, as a list

June 2025, Core Pure, Q5 (4 marks): - 5 The cubic equation x 2 3 x 4 0 has roots a, b and c. + = 1 a + 1 b + 1 c + Determine a cubic equation with integer coefficients whose roots are ( 1 ), ( 1 )
June 2025, Core Pure, Q12 (8 marks): In this question you must show detailed reasoning. are a, 2 4 3 2 The roots of the equation z z cz dz 18 0 a , b and - . - + + + = b Determine, in any order, th
June 2025, AS Core Pure, Q8 (8 marks): 2 8 The three distinct roots of the equation z 4 z pz q 0 , where p and q are real, are drawn on - + + = an Argand diagram. The three points which represent the
June 2024, Core Pure, Q3 (4 marks): 2 3 The equation x 2 2 x 8 x 15 0 has roots a, b and c. - + - = 2 2 2 Determine the value of . [4] a + b + c
June 2024, AS Core Pure, Q4 (7 marks): In this question you must show detailed reasoning. 3 2 The roots of the cubic equation x 3 x 19 x 17 0 are a, b and c. - + - = 1 a - 1 b - (a) Find a cubic equa
June 2023, Core Pure, Q10 (7 marks): 2 10 The equation x 4 x 7 x c 0 , where c is a constant, has roots a, b and a + . - + + = b (a) Determine the roots of the equation. [6] (b) Find c. [1]
June 2023, AS Core Pure, Q2 (4 marks): In this question you must show detailed reasoning. 2 - The equation x kx 2 k 0 , where k is a non-zero constant, has roots a and b. + = Find a b + a in terms of
June 2022, Core Pure, Q10 (10 marks): The equation 4 3 2 4 x 16 x ax bx 6 0 , + + + + = where a and b are real, has roots a, 2 a , b and 3b. (a) Given that ,0 determine all 4 roots. [6] 1 b (b) Dete
June 2022, AS Core Pure, Q4 (6 marks): In this question you must show detailed reasoning. where k is a constant, has roots a, 1 3 2 z z kz The equation 2 3 0 , a and b. + + + = [6] Determine the root
November 2021, Core Pure, Q8 (9 marks): , where p and q are constants, has roots a, -a, b and 1 4 3 2 8 The equation x 4 4 x px qx 9 0 b . - + + - = (a) Determine the exact roots of the equation. [5]
November 2021, AS Core Pure, Q2 (3 marks): - 2 The equation x 3 4 x 2 0 has roots a and b. + = Find an equation with integer coefficients whose roots are 3 - 2a and 3 - 2b. [3]
November 2021, AS Core Pure, Q8 (7 marks): In this question you must show detailed reasoning. a . Given that 3 2 The equation x kx 15 x 25 0 has roots a, b and b 2 0 , find, in any order, + + - = a • the
November 2020, Core Pure, Q4 (8 marks): - 4 The roots of the equation x 2 5 x 7 0 are a, b and c. + = (a) Find 1 1 1 . [4] + + a c b (b) Find an equation with integer coefficients whose roots are 2a -
November 2020, AS Core Pure, Q7 (7 marks): 3 2 7 In the quartic equation x 2 20 x ax bx 250 0 , the coefficients a and b are real. One root - + + + = of the equation is 2 + . i Find the other roots. [7]
June 2019, Core Pure, Q8 (8 marks): In this question you must show detailed reasoning. 1 3 2 The roots of the equation x x kx 2 0 are , and b. - + - = a a (a) Evaluate, in exact form, the roots of
June 2019, AS Core Pure, Q2 (3 marks): - + 2 The roots of the equation x 3 x 2 0 are a and b. = Find a quadratic equation with integer coefficients whose roots are 2 a - and 2 3 b - . 3 [3]
June 2018, AS Core Pure, Q4 (5 marks): [5] Find a cubic equation with real coefficients, two of whose roots are 2 - i and 3.

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