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Further algebra and functions: OCR MEI A‑Level Further Maths questions

Every OCR MEI A-Level Further Maths exam question on CP.4 Further algebra and functions, 2018–2025, with its mark scheme and examiner’s report.

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847 questions

  1. Q3

    June 2025 · Core Pure

    5 marksCP.4 Further algebra and functions
    The whole paper ↗
    June 2025 Core Pure, question 3: Using standard summation formulae, show that, for integers n H , 1 1 1 3 2 4 ... n ( n 2 ) n n ( 1 )( an b ), # + # + + # + = + + 6 where a and b are integers t

  2. Q5

    June 2025 · Core Pure

    4 marksCP.4 Further algebra and functions
    The whole paper ↗
    June 2025 Core Pure, question 5: - 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 )

  3. Q8

    June 2025 · Core Pure

    7 marksCP.1 Proof by inductionCP.4 Further algebra and functions
    The whole paper ↗
    June 2025 Core Pure, question 8: The function ( )x f is defined as ( ) f x ln ( 1 x ) , for x 2- . 1 = + f ( ) n (a) Prove by mathematical induction that the nth derivative of ( ) f x , ( )x ,

  4. Q12

    June 2025 · Core Pure

    8 marksCP.4 Further algebra and functionsCP.2 Complex numbers
    The whole paper ↗
    June 2025 Core Pure, question 12: 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

  5. Q6

    June 2025 · AS Core Pure

    7 marksCP.4 Further algebra and functions
    The whole paper ↗
    June 2025 AS Core Pure, question 6: 1 6 (a) Express ( as a single simplified fraction. [2] - 2 2 r 1 ) ( r 1 ) - + n r =/ (b) Hence determine the limit which converges to as n " 3. [5] 2 2 ( r 1 )

  6. Q8

    June 2025 · AS Core Pure

    8 marksCP.2 Complex numbersCP.4 Further algebra and functions
    The whole paper ↗
    June 2025 AS Core Pure, question 8: 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

  7. Q1

    June 2024 · Core Pure

    4 marksCP.4 Further algebra and functions
    The whole paper ↗
    June 2024 Core Pure, question 1: r/ 1 1 1 1 By expressing r - + as a single fraction, find in terms of n. [4] 1 ( r 1 )( r 2 ) 1 r 2 + + + =

  8. Q3

    June 2024 · Core Pure

    4 marksCP.4 Further algebra and functions
    The whole paper ↗
    June 2024 Core Pure, question 3: 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

  9. Q10

    June 2024 · Core Pure

    6 marksCP.4 Further algebra and functions
    The whole paper ↗
    June 2024 Core Pure, question 10: 10 (a) Write down the first three terms of the Maclaurin series for ln ( 1 x ) . [1] + n (b) Use these three terms to show that ln ( . 1 125 ) , where n is an i

  10. Q3

    June 2024 · AS Core Pure

    6 marksCP.4 Further algebra and functions
    The whole paper ↗
    June 2024 AS Core Pure, question 3: n =/ 3 3 (a) Using standard summation formulae, write down an expression in terms of n for r . [1] r 1 2 n = +/ 3 1 2 (b) Hence show that r n ( an b cn )( d ) ,

  11. Q4

    June 2024 · AS Core Pure

    7 marksCP.4 Further algebra and functions
    The whole paper ↗
    June 2024 AS Core Pure, question 4: 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

  12. Q3

    June 2023 · Core Pure

    6 marksCP.4 Further algebra and functions
    The whole paper ↗
    June 2023 Core Pure, question 3: (a) Using partial fractions and the method of differences, show that 1 1 1 1 3 an b + … , + + + + = - 1 3 2 4 3 5 n n ( 2 ) 4 2 ( n 1 )( n 2 ) + + + # # # where

  13. Q4

    June 2023 · Core Pure

    6 marksCP.4 Further algebra and functions
    The whole paper ↗
    June 2023 Core Pure, question 4: (a) (i) Given that ( )x f 1 2 x , find and . [2] f l ( )x f m ( )x = + (ii) Hence, find the first three terms of the Maclaurin series for 1 2 x . [2] + 143 (b)

  14. Q10

    June 2023 · Core Pure

    7 marksCP.4 Further algebra and functions
    The whole paper ↗
    June 2023 Core Pure, question 10: 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]

  15. Q2

    June 2023 · AS Core Pure

    4 marksCP.4 Further algebra and functions
    The whole paper ↗
    June 2023 AS Core Pure, question 2: 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

  16. Q3

    June 2023 · AS Core Pure

    6 marksCP.4 Further algebra and functionsCP.2 Complex numbers
    The whole paper ↗
    June 2023 AS Core Pure, question 3: In this question you must show detailed reasoning. 3 2 The function f (z) is given by ( ) f z 2 z 7 z 16 z 15 . = - + - 3a k, find the roots of f (z) = 0. By fi

  17. Q4

    June 2023 · AS Core Pure

    6 marksCP.4 Further algebra and functions
    The whole paper ↗
    June 2023 AS Core Pure, question 4: n =/ 2 4 You are given that ( ar b ) n for all n, where a and b are constants. + = r 1 n =/ By finding ( ar b ) in terms of a, b and n, determine the values of

  18. Q1

    June 2022 · Core Pure

    7 marksCP.4 Further algebra and functions
    The whole paper ↗
    June 2022 Core Pure, question 1: n =/ 1 3 3 2 1 (a) By considering ( r ) r , find ( 3 r 3 r 1 ). [3] + - + + r 1 n =/ (b) Use this result to find r r ( 1 ), expressing your answer in fully fact

  19. Q9

    June 2022 · Core Pure

    12 marksCP.4 Further algebra and functionsCP.8 Hyperbolic functions
    The whole paper ↗
    June 2022 Core Pure, question 9: The function ( )x f is defined by ( ) f x ln ( 1 sinh x ). = + (a) Given that k lies in the domain of this function, explain why k must be greater than ln ( 2 1

  20. Q10

    June 2022 · Core Pure

    10 marksCP.4 Further algebra and functionsCP.2 Complex numbers
    The whole paper ↗
    June 2022 Core Pure, question 10: 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
All 41 questions on Further algebra and functions, as a list
  1. June 2025, Core Pure, Q3 (5 marks): Using standard summation formulae, show that, for integers n H , 1 1 1 3 2 4 ... n ( n 2 ) n n ( 1 )( an b ), # + # + + # + = + + 6 where a and b are integers t
  2. 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 )
  3. June 2025, Core Pure, Q8 (7 marks): The function ( )x f is defined as ( ) f x ln ( 1 x ) , for x 2- . 1 = + f ( ) n (a) Prove by mathematical induction that the nth derivative of ( ) f x , ( )x ,
  4. 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
  5. June 2025, AS Core Pure, Q6 (7 marks): 1 6 (a) Express ( as a single simplified fraction. [2] - 2 2 r 1 ) ( r 1 ) - + n r =/ (b) Hence determine the limit which converges to as n " 3. [5] 2 2 ( r 1 )
  6. 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
  7. June 2024, Core Pure, Q1 (4 marks): r/ 1 1 1 1 By expressing r - + as a single fraction, find in terms of n. [4] 1 ( r 1 )( r 2 ) 1 r 2 + + + =
  8. 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
  9. June 2024, Core Pure, Q10 (6 marks): 10 (a) Write down the first three terms of the Maclaurin series for ln ( 1 x ) . [1] + n (b) Use these three terms to show that ln ( . 1 125 ) , where n is an i
  10. June 2024, AS Core Pure, Q3 (6 marks): n =/ 3 3 (a) Using standard summation formulae, write down an expression in terms of n for r . [1] r 1 2 n = +/ 3 1 2 (b) Hence show that r n ( an b cn )( d ) ,
  11. 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
  12. June 2023, Core Pure, Q3 (6 marks): (a) Using partial fractions and the method of differences, show that 1 1 1 1 3 an b + … , + + + + = - 1 3 2 4 3 5 n n ( 2 ) 4 2 ( n 1 )( n 2 ) + + + # # # where
  13. June 2023, Core Pure, Q4 (6 marks): (a) (i) Given that ( )x f 1 2 x , find and . [2] f l ( )x f m ( )x = + (ii) Hence, find the first three terms of the Maclaurin series for 1 2 x . [2] + 143 (b)
  14. 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]
  15. 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
  16. June 2023, AS Core Pure, Q3 (6 marks): In this question you must show detailed reasoning. 3 2 The function f (z) is given by ( ) f z 2 z 7 z 16 z 15 . = - + - 3a k, find the roots of f (z) = 0. By fi
  17. June 2023, AS Core Pure, Q4 (6 marks): n =/ 2 4 You are given that ( ar b ) n for all n, where a and b are constants. + = r 1 n =/ By finding ( ar b ) in terms of a, b and n, determine the values of
  18. June 2022, Core Pure, Q1 (7 marks): n =/ 1 3 3 2 1 (a) By considering ( r ) r , find ( 3 r 3 r 1 ). [3] + - + + r 1 n =/ (b) Use this result to find r r ( 1 ), expressing your answer in fully fact
  19. June 2022, Core Pure, Q9 (12 marks): The function ( )x f is defined by ( ) f x ln ( 1 sinh x ). = + (a) Given that k lies in the domain of this function, explain why k must be greater than ln ( 2 1
  20. 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
  21. 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
  22. June 2022, AS Core Pure, Q6 (10 marks): n =/ 1 6 (a) Using standard summation formulae, show that r r ( 2 ) n n ( 1 2 )( n 7 ). [4] + = + + 6 r 1 (b) Use induction to prove the result in part (a). [6]
  23. November 2021, Core Pure, Q1 (7 marks): 1 (a) Express ( in partial fractions. [3] 2 r 1 2 )( r 1 ) - + n r/ 1 (b) Hence find , expressing the result as a single fraction. [4] = ( 2 r 1 2 )( r 1 ) - +
  24. November 2021, Core Pure, Q5 (6 marks): (a) Use a Maclaurin series to find a quadratic approximation for ln ( 1 2 x ) . [1] + (b) Find the percentage error in using the approximation in part (a) to ca
  25. 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]
  26. November 2021, AS Core Pure, Q1 (3 marks): n ^ h / 2 - 1 Using standard summation formulae, find r 3 r , giving your answer in fully factorised form. [3] r 1 =
  27. 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]
  28. 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
  29. November 2020, Core Pure, Q1 (6 marks): Using standard summation of series formulae, determine the sum of the first n terms of the series ( 1 2 4 ) ( 2 3 5 ) ( 3 4 6 ) + …, + + # # # # # # where n is
  30. 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 -
  31. November 2020, Core Pure, Q13 (9 marks): (a) Using exponentials, prove that sinh 2 x 2 cosh sinh x x . [2] = 2 (b) Hence show that if ( ) f x sinh x , then f ″( ) x 2 cosh 2 x . [2] = = 2 are all zero.
  32. November 2020, AS Core Pure, Q1 (3 marks): In this question you must show detailed reasoning. J N 50 1 / 1 KK OO Find , expressing the answer as an exact fraction. [3] - + r 1 r 1 - r L P 2 =
  33. 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]
  34. June 2019, Core Pure, Q1 (4 marks): ^ h n / 2 1 Find 2 r 1 , expressing your answer in fully factorised form. [4] - r 1 =
  35. June 2019, Core Pure, Q5 (5 marks): Using the Maclaurin series for cos x 2 , show that, for small values of x, 2 2 4 6 sin x ax bx cx , . + + where the values of a, b and c are to be given in exac
  36. 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
  37. June 2019, AS Core Pure, Q1 (3 marks): In this question you must show detailed reasoning. 1J N 100 1 / KK OO Find , giving your answer correct to 4 decimal places. [3] - + r r 2 r 1 L P =
  38. 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]
  39. June 2019, AS Core Pure, Q8 (11 marks): In this question you must show detailed reasoning. 4 3 2 You are given that i is a root of the equation z 2 z 3 z az b 0 , where a and b are real - + + + = cons
  40. 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.
  41. June 2018, AS Core Pure, Q7 (9 marks): 1 7 (i) Express r [2] + as a single fraction. - r 2 1 2 1 - (ii) Find how many terms of the series 2 2 2 2 + + + f + + f 1 3 3 5 5 7 ( 2 r 1 2 )( r 1 ) - + # #

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