Skip to content
Try on iPad

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

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

Find questions648

648 questions

  1. Q4

    June 2025 · Pure Core 1

    9 marksCP.4 Further algebra and functions
    The whole paper ↗
    June 2025 Pure Core 1, question 4: 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 ( ,

  2. Q6

    June 2025 · Pure Core 1

    8 marksCP.4 Further algebra and functions
    The whole paper ↗
    June 2025 Pure Core 1, question 6: In this question you must show detailed reasoning. J N J N J N 1 1 1 1 1 1 KK OO KK OO KK OO for n The series Sn is given by S +. = + + + Z ! # # g # n n n 5 15

  3. Q4

    June 2025 · Pure Core 2

    4 marksCP.4 Further algebra and functions
    The whole paper ↗
    June 2025 Pure Core 2, question 4: In this question you must show detailed reasoning. Determine the sum of all cube numbers from 216 to 512 000 inclusive. [4]

  4. Q8

    June 2025 · Pure Core 2

    12 marksCP.1 Proof by inductionCP.4 Further algebra and functions
    The whole paper ↗
    June 2025 Pure Core 2, question 8: A function f (x) is defined by f (x) = xsinh 2x. 2 n 0 d f d f n (a) Prove by induction that 4 ( x sinh 2 x n cosh 2 x ) for n 0 where d is defined as H = + 2 n

  5. Q3

    June 2025 · AS Pure Core

    6 marksCP.4 Further algebra and functions
    The whole paper ↗
    June 2025 AS Pure Core, question 3: + 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 (

  6. Q4

    June 2024 · Pure Core 1

    3 marksCP.4 Further algebra and functions
    The whole paper ↗
    June 2024 Pure Core 1, question 4: 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

  7. Q9

    June 2024 · Pure Core 1

    6 marksCP.4 Further algebra and functionsCP.5 Further calculus
    The whole paper ↗
    June 2024 Pure Core 1, question 9: ^ h up to and including the term in x4. 2 9 (a) Find the Maclaurin series of x [3] ln ( 1 ) + ^ h and 2 y x The diagram below shows parts of the graphs of the c

  8. Q1

    June 2024 · Pure Core 2

    5 marksCP.4 Further algebra and functions
    The whole paper ↗
    June 2024 Pure Core 2, question 1: J N n / 1 1 1 KK OO 1 (a) Use the method of differences to show that 1 . [1] - + = - n 1 r r 1 + L P r 1 = (b) Hence determine the following sums. 99 / 1 1 (i)

  9. Q4

    June 2024 · Pure Core 2

    5 marksCP.4 Further algebra and functions
    The whole paper ↗
    June 2024 Pure Core 2, question 4: In this question you must show detailed reasoning. The series S is defined as being the sum of the squares of all positive odd integers from 12 to 7792. Determi

  10. Q7

    June 2024 · AS Pure Core

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

  11. Q1

    June 2023 · Pure Core 1

    3 marksCP.4 Further algebra and functions
    The whole paper ↗
    June 2023 Pure Core 1, question 1: In this question you must show detailed reasoning. 50 =/ 2 Determine the value of r ( 16 r ) . [3] - r 1

  12. Q2

    June 2023 · Pure Core 1

    6 marksCP.4 Further algebra and functions
    The whole paper ↗
    June 2023 Pure Core 1, question 2: 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

  13. Q7

    June 2023 · Pure Core 2

    8 marksCP.4 Further algebra and functions
    The whole paper ↗
    June 2023 Pure Core 2, question 7: In this question you must show detailed reasoning. (a) Show that n n / / r a 5 6 1 + b c = + + n 3 2 1 2 r r + r + r 1 r 1 = = [6] where a, b and c are integers

  14. Q9

    June 2023 · Pure Core 2

    9 marksCP.1 Proof by inductionCP.4 Further algebra and functions
    The whole paper ↗
    June 2023 Pure Core 2, question 9: A function is defined by y f ( ) t where ( ) f t ln ( 1 at ) and a is a constant. = = + 2 3 4 n d , t y d y d y d y d y (a) By considering t , t and t , make a

  15. Q1

    June 2023 · AS Pure Core

    5 marksCP.4 Further algebra and functions
    The whole paper ↗
    June 2023 AS Pure Core, question 1: 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 + ,

  16. Q5

    June 2023 · AS Pure Core

    4 marksCP.4 Further algebra and functions
    The whole paper ↗
    June 2023 AS Pure Core, question 5: 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

  17. Q3

    June 2022 · Pure Core 2

    6 marksCP.4 Further algebra and functions
    The whole paper ↗
    June 2022 Pure Core 2, question 3: 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

  18. Q4

    June 2022 · Pure Core 2

    4 marksCP.4 Further algebra and functions
    The whole paper ↗
    June 2022 Pure Core 2, question 4: In this question you must show detailed reasoning. 2 2 2 1 2 ... n + + + Determine the smallest value of n for which 2 341 . [4] n 1 2 ... + + +

  19. Q8

    June 2022 · Pure Core 2

    7 marksCP.4 Further algebra and functions
    The whole paper ↗
    June 2022 Pure Core 2, question 8: In this question you must show detailed reasoning. 98 5 r 2 20539 =/ + It is given that for some k. = r r ( 1 )( r 2 ) 34650 + + r k Determine the value of k. [

  20. Q3

    June 2022 · AS Pure Core

    6 marksCP.4 Further algebra and functions
    The whole paper ↗
    June 2022 AS Pure Core, question 3: 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
All 38 questions on Further algebra and functions, as a list
  1. 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 ( ,
  2. June 2025, Pure Core 1, Q6 (8 marks): In this question you must show detailed reasoning. J N J N J N 1 1 1 1 1 1 KK OO KK OO KK OO for n The series Sn is given by S +. = + + + Z ! # # g # n n n 5 15
  3. June 2025, Pure Core 2, Q4 (4 marks): In this question you must show detailed reasoning. Determine the sum of all cube numbers from 216 to 512 000 inclusive. [4]
  4. June 2025, Pure Core 2, Q8 (12 marks): A function f (x) is defined by f (x) = xsinh 2x. 2 n 0 d f d f n (a) Prove by induction that 4 ( x sinh 2 x n cosh 2 x ) for n 0 where d is defined as H = + 2 n
  5. 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 (
  6. 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
  7. June 2024, Pure Core 1, Q9 (6 marks): ^ h up to and including the term in x4. 2 9 (a) Find the Maclaurin series of x [3] ln ( 1 ) + ^ h and 2 y x The diagram below shows parts of the graphs of the c
  8. June 2024, Pure Core 2, Q1 (5 marks): J N n / 1 1 1 KK OO 1 (a) Use the method of differences to show that 1 . [1] - + = - n 1 r r 1 + L P r 1 = (b) Hence determine the following sums. 99 / 1 1 (i)
  9. June 2024, Pure Core 2, Q4 (5 marks): In this question you must show detailed reasoning. The series S is defined as being the sum of the squares of all positive odd integers from 12 to 7792. Determi
  10. 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
  11. June 2023, Pure Core 1, Q1 (3 marks): In this question you must show detailed reasoning. 50 =/ 2 Determine the value of r ( 16 r ) . [3] - r 1
  12. 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
  13. June 2023, Pure Core 2, Q7 (8 marks): In this question you must show detailed reasoning. (a) Show that n n / / r a 5 6 1 + b c = + + n 3 2 1 2 r r + r + r 1 r 1 = = [6] where a, b and c are integers
  14. June 2023, Pure Core 2, Q9 (9 marks): A function is defined by y f ( ) t where ( ) f t ln ( 1 at ) and a is a constant. = = + 2 3 4 n d , t y d y d y d y d y (a) By considering t , t and t , make a
  15. 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 + ,
  16. 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
  17. 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
  18. June 2022, Pure Core 2, Q4 (4 marks): In this question you must show detailed reasoning. 2 2 2 1 2 ... n + + + Determine the smallest value of n for which 2 341 . [4] n 1 2 ... + + +
  19. June 2022, Pure Core 2, Q8 (7 marks): In this question you must show detailed reasoning. 98 5 r 2 20539 =/ + It is given that for some k. = r r ( 1 )( r 2 ) 34650 + + r k Determine the value of k. [
  20. 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
  21. November 2021, Pure Core 1, Q2 (8 marks): 2 You are given that ( ) f x tan - ( 1 x ) . = + (a) (i) Find the value of ( ) f 0 . [1] (ii) Determine the value of f l ( ) 0 . [2] 1 (iii) Show that f ( ) 0 .
  22. 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,
  23. November 2021, Pure Core 1, Q10 (8 marks): n =/ 1 10 Using an algebraic method, determine the least value of n for which 0 49 . . [8] H r r ( 2 1 2 )( 1 ) - + r 1
  24. November 2021, Pure Core 2, Q4 (3 marks): In this question you must show detailed reasoning. 100 =/ 2 Determine the value of ( 2 r 3 ) . [3] + r 1
  25. November 2021, Pure Core 2, Q10 (6 marks): In this question you must show detailed reasoning. ex 2 (a) By using an appropriate Maclaurin series prove that if x 2 0 then 1 + . x [2] t 2 (b) Hence, by usin
  26. 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 +
  27. November 2020, Pure Core 1, Q2 (3 marks): ^ h in terms of n. Give your answer in 2 2 2 2 2 Find an expression for 1 2 2 3 3 4 n n 1 + + + f + + # # # fully factorised form. [3] J N 1 0 0
  28. 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
  29. 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
  30. November 2020, Pure Core 2, Q3 (6 marks): In this question you must show detailed reasoning. n =/ 3 37 1 1 1 (a) Use partial fractions to show that . [5] = - - - n 60 n 1 n 2 2 r r + + 2 + - r 5 n / f 3
  31. November 2020, Pure Core 2, Q10 (10 marks): 10 Let ( ) f x sin - ( ) x . = (a) (i) Determine f ll ( )x . [2] (ii) Determine the first two non-zero terms of the Maclaurin expansion for ( )x f . [3] (iii) B
  32. 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
  33. 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
  34. June 2019, Pure Core 1, Q4 (3 marks): ^ h n n 10 =/ and =/ 2 / 4 Using the formulae for r r , show that r 3 r 2 1045 . [3] - = r 1 r 1 r 1 =
  35. June 2019, Pure Core 2, Q1 (7 marks): In this question you must show detailed reasoning. n 1 1 1 =/ (a) By using partial fractions show that . [5] = - 2 n 2 2 r r + 3 2 + + r 1 1 3 =/ (b) Hence dete
  36. June 2019, Pure Core 2, Q10 (7 marks): (a) Use differentiation to find the first two non-zero terms of the Maclaurin expansion of c m. 1 + ln cos x [4] 2 c m c m. 1 3 (b) By considering the root of t
  37. 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
  38. 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 + - +

Questions, mark schemes and examiner’s reports are © OCR, from their published past papers. Plumi isn’t affiliated with any exam board. Want whole papers? Every OCR A-Level Further Maths paper.

Rather write your answers by hand on iPad? Plumi marks them against the same mark schemes.