Skip to content
Try on iPad

Matrices: OCR MEI A‑Level Further Maths questions

Every OCR MEI A-Level Further Maths exam question on CP.3 Matrices, 2018–2025, with its mark scheme and examiner’s report.

Find questions847

847 questions

  1. Q4

    June 2025 · Core Pure

    7 marksCP.3 Matrices
    The whole paper ↗
    June 2025 Core Pure, question 4: (a) You are given that M and N are non-singular 2 # matrices. 2 Write down the product rule for the inverse matrices of M, N and MN. [1] (b) Verify this rule fo

  2. Q14

    June 2025 · Core Pure

    10 marksCP.8 Hyperbolic functionsCP.3 Matrices
    The whole paper ↗
    June 2025 Core Pure, question 14: - , show that 14 (a) By using the definition of cosh x and sinh x in terms of ex and e x 2 2 x x x [2] cosh sinh cosh 2 . + / J cosh x sinh x N KKK OOO and (b)

  3. Q1

    June 2025 · AS Core Pure

    5 marksCP.3 Matrices
    The whole paper ↗
    June 2025 AS Core Pure, question 1: The matrices A and B are given by J N J N 1 3 2 1 KK OO and B KK OO. A = = 0 1 0 1 L P L P (a) Use A and B to show that matrix multiplication is not, in general

  4. Q3

    June 2025 · AS Core Pure

    7 marksCP.3 Matrices
    The whole paper ↗
    June 2025 AS Core Pure, question 3: The matrices M and N are given by J N J N a b b a - - KK OO and KK OO where a and b are positive constants. M = N = b a a b L P L P 2 = (a) Given that M N , det

  5. Q4

    June 2025 · AS Core Pure

    8 marksCP.3 Matrices
    The whole paper ↗
    June 2025 AS Core Pure, question 4: K O 4 (a) The transformation T is represented by the matrix M 2 1 0 OO. = KK 1 2 1 - L P A shape S1 is mapped to a shape S2 by the transformation T. Show that v

  6. Q8

    June 2024 · Core Pure

    10 marksCP.3 MatricesCP.1 Proof by induction
    The whole paper ↗
    June 2024 Core Pure, question 8: (a) Specify fully the transformation T of the plane associated with the matrix M, where = c 1 m and m is a non-zero constant. m M [2] 0 1 (b) (i) Find det M. [1

  7. Q15

    June 2024 · Core Pure

    10 marksCP.3 Matrices
    The whole paper ↗
    June 2024 Core Pure, question 15: Three planes have equations x ky 3 z = , 1 + + 3 x 4 y 2 z = , 3 + + x 3 y z =- , k + - where k is a constant. (a) Show that the planes meet at a point except f

  8. Q2

    June 2024 · AS Core Pure

    5 marksCP.3 Matrices
    The whole paper ↗
    June 2024 AS Core Pure, question 2: J N J N J N 1 a 2 0 1 0 = - KK OO, B KK OO and C KK OO, where a is 2 The matrices A, B and C are given by A = - = 1 2 1 1 2 1 - a constant. L P L P L P ^ h ^ h

  9. Q5

    June 2024 · AS Core Pure

    6 marksCP.3 Matrices
    The whole paper ↗
    June 2024 AS Core Pure, question 5: K O 5 (a) Find the volume scale factor of the transformation with associated matrix 0 3 1 OO. [2] - KK 1 0 2 - L P (b) The transformations S and T of the plane

  10. Q6

    June 2024 · AS Core Pure

    9 marksCP.1 Proof by inductionCP.3 Matrices
    The whole paper ↗
    June 2024 AS Core Pure, question 6: J N 4 9 - KK OO. 6 You are given that M = 1 2 - L P J N n n 1 3 9 + - n = KK OO for all positive integers n. (a) Prove that M [6] n 1 3 n - L P (b) A student th

  11. Q7

    June 2024 · AS Core Pure

    6 marksCP.3 Matrices
    The whole paper ↗
    June 2024 AS Core Pure, question 7: Three planes have equations x 2 y 3 z 0 , + - = x 3 y 2 z 0 , - + - = x 2 y k z k , - + = where k is a constant. (a) For the case k = 0, the origin lies on all

  12. Q6

    June 2023 · Core Pure

    9 marksCP.3 Matrices
    The whole paper ↗
    June 2023 Core Pure, question 6: J N J N The matrices M and N are 0 1 OO and 2 0 KK KK OO respectively. 6 1 0 0 1 L P L P (a) In this question you must show detailed reasoning. Determine whethe

  13. Q14

    June 2023 · Core Pure

    13 marksCP.3 Matrices
    The whole paper ↗
    June 2023 Core Pure, question 14: Three planes have equations kx - z 2 , = x ky 2 z = , 1 - + + 2 kx 2 y 3 z 0 , + + = where k is a constant. (a) By considering a suitable determinant, show that

  14. Q1

    June 2023 · AS Core Pure

    3 marksCP.3 Matrices
    The whole paper ↗
    June 2023 AS Core Pure, question 1: The transformation R of the plane is reflection in the line x = 0. (a) Write down the matrix M associated with R. [1] (b) Find M2. [1] (c) Interpret the result

  15. Q6

    June 2023 · AS Core Pure

    8 marksCP.3 MatricesCP.1 Proof by induction
    The whole paper ↗
    June 2023 AS Core Pure, question 6: J N 2 1 KK OO. 6 The matrix M is 1 0 - L P (a) Calculate M2, M3 and M4. [2] (b) Hence make a conjecture about the matrix Mn. [1] (c) Prove your conjecture. [5]

  16. Q8

    June 2023 · AS Core Pure

    6 marksCP.3 Matrices
    The whole paper ↗
    June 2023 AS Core Pure, question 8: The equations of three planes are 2 x y 3 z 3 , + + = 3 x y 2 z 2 , - - = 4 x 3 y 7 z k , - + + = where k is a constant. (a) By considering a suitable determina

  17. Q9

    June 2023 · AS Core Pure

    6 marksCP.3 Matrices
    The whole paper ↗
    June 2023 AS Core Pure, question 9: J N k 1 1 + - KK OO, where k is a 9 A transformation T of the plane is represented by the matrix M = 1 k constant. L P Show that, for all values of k, T has no

  18. Q4

    June 2022 · Core Pure

    7 marksCP.3 Matrices
    The whole paper ↗
    June 2022 Core Pure, question 4: K O 4 (a) A transformation with associated matrix 0 1 2 OO, where m is a constant, maps the vertices KK - 2 0 3 L P of a cube to points that all lie in a plane.

  19. Q6

    June 2022 · Core Pure

    5 marksCP.1 Proof by inductionCP.3 Matrices
    The whole paper ↗
    June 2022 Core Pure, question 6: J N J N n 2 0 2 0 KK OO KK OO for all positive integers n. 6 Prove by mathematical induction that [5] = n 1 1 1 2 1 - - L P L P 3 Answer all the questions.

  20. Q1

    June 2022 · AS Core Pure

    6 marksCP.3 Matrices
    The whole paper ↗
    June 2022 AS Core Pure, question 1: (a) (i) Write the following simultaneous equations as a matrix equation. x y 2 z 7 + + = 2 x 4 y 3 z 5 - - =- [1] 5 x 3 y 5 z 13 - + + = (ii) Hence solve the eq
All 45 questions on Matrices, as a list
  1. June 2025, Core Pure, Q4 (7 marks): (a) You are given that M and N are non-singular 2 # matrices. 2 Write down the product rule for the inverse matrices of M, N and MN. [1] (b) Verify this rule fo
  2. June 2025, Core Pure, Q14 (10 marks): - , show that 14 (a) By using the definition of cosh x and sinh x in terms of ex and e x 2 2 x x x [2] cosh sinh cosh 2 . + / J cosh x sinh x N KKK OOO and (b)
  3. June 2025, AS Core Pure, Q1 (5 marks): The matrices A and B are given by J N J N 1 3 2 1 KK OO and B KK OO. A = = 0 1 0 1 L P L P (a) Use A and B to show that matrix multiplication is not, in general
  4. June 2025, AS Core Pure, Q3 (7 marks): The matrices M and N are given by J N J N a b b a - - KK OO and KK OO where a and b are positive constants. M = N = b a a b L P L P 2 = (a) Given that M N , det
  5. June 2025, AS Core Pure, Q4 (8 marks): K O 4 (a) The transformation T is represented by the matrix M 2 1 0 OO. = KK 1 2 1 - L P A shape S1 is mapped to a shape S2 by the transformation T. Show that v
  6. June 2024, Core Pure, Q8 (10 marks): (a) Specify fully the transformation T of the plane associated with the matrix M, where = c 1 m and m is a non-zero constant. m M [2] 0 1 (b) (i) Find det M. [1
  7. June 2024, Core Pure, Q15 (10 marks): Three planes have equations x ky 3 z = , 1 + + 3 x 4 y 2 z = , 3 + + x 3 y z =- , k + - where k is a constant. (a) Show that the planes meet at a point except f
  8. June 2024, AS Core Pure, Q2 (5 marks): J N J N J N 1 a 2 0 1 0 = - KK OO, B KK OO and C KK OO, where a is 2 The matrices A, B and C are given by A = - = 1 2 1 1 2 1 - a constant. L P L P L P ^ h ^ h
  9. June 2024, AS Core Pure, Q5 (6 marks): K O 5 (a) Find the volume scale factor of the transformation with associated matrix 0 3 1 OO. [2] - KK 1 0 2 - L P (b) The transformations S and T of the plane
  10. June 2024, AS Core Pure, Q6 (9 marks): J N 4 9 - KK OO. 6 You are given that M = 1 2 - L P J N n n 1 3 9 + - n = KK OO for all positive integers n. (a) Prove that M [6] n 1 3 n - L P (b) A student th
  11. June 2024, AS Core Pure, Q7 (6 marks): Three planes have equations x 2 y 3 z 0 , + - = x 3 y 2 z 0 , - + - = x 2 y k z k , - + = where k is a constant. (a) For the case k = 0, the origin lies on all
  12. June 2023, Core Pure, Q6 (9 marks): J N J N The matrices M and N are 0 1 OO and 2 0 KK KK OO respectively. 6 1 0 0 1 L P L P (a) In this question you must show detailed reasoning. Determine whethe
  13. June 2023, Core Pure, Q14 (13 marks): Three planes have equations kx - z 2 , = x ky 2 z = , 1 - + + 2 kx 2 y 3 z 0 , + + = where k is a constant. (a) By considering a suitable determinant, show that
  14. June 2023, AS Core Pure, Q1 (3 marks): The transformation R of the plane is reflection in the line x = 0. (a) Write down the matrix M associated with R. [1] (b) Find M2. [1] (c) Interpret the result
  15. June 2023, AS Core Pure, Q6 (8 marks): J N 2 1 KK OO. 6 The matrix M is 1 0 - L P (a) Calculate M2, M3 and M4. [2] (b) Hence make a conjecture about the matrix Mn. [1] (c) Prove your conjecture. [5]
  16. June 2023, AS Core Pure, Q8 (6 marks): The equations of three planes are 2 x y 3 z 3 , + + = 3 x y 2 z 2 , - - = 4 x 3 y 7 z k , - + + = where k is a constant. (a) By considering a suitable determina
  17. June 2023, AS Core Pure, Q9 (6 marks): J N k 1 1 + - KK OO, where k is a 9 A transformation T of the plane is represented by the matrix M = 1 k constant. L P Show that, for all values of k, T has no
  18. June 2022, Core Pure, Q4 (7 marks): K O 4 (a) A transformation with associated matrix 0 1 2 OO, where m is a constant, maps the vertices KK - 2 0 3 L P of a cube to points that all lie in a plane.
  19. June 2022, Core Pure, Q6 (5 marks): J N J N n 2 0 2 0 KK OO KK OO for all positive integers n. 6 Prove by mathematical induction that [5] = n 1 1 1 2 1 - - L P L P 3 Answer all the questions.
  20. June 2022, AS Core Pure, Q1 (6 marks): (a) (i) Write the following simultaneous equations as a matrix equation. x y 2 z 7 + + = 2 x 4 y 3 z 5 - - =- [1] 5 x 3 y 5 z 13 - + + = (ii) Hence solve the eq
  21. June 2022, AS Core Pure, Q8 (12 marks): J N cos 2 cos sin - i i i KK OO. 8 A transformation T of the plane has matrix M, where M = sin 2 sin cos + i i i L P (a) Show that T leaves areas unchanged for
  22. November 2021, Core Pure, Q6 (4 marks): J N is an invariant line of the transformation with matrix 1 2 KK OO, determine the 6 Given that y mx = 2 2 - L P possible values of m. [4]
  23. November 2021, Core Pure, Q9 (11 marks): J N 1 0 = - KK OO. 9 The transformation T of the plane has associated matrix M, where M 2 1 - L P (a) On the grid in the Printed Answer Booklet, plot the image
  24. November 2021, Core Pure, Q15 (6 marks): The equations of three planes are 4 x ky 7 z 4 , - + + = x 2 y 5 z l , - + = 2 x 3 y z 2 . + + = Given that the planes form a sheaf, determine the values of k a
  25. November 2021, AS Core Pure, Q3 (7 marks): Three planes have the following equations. 2 x 3 y z 3 , - + =- x 4 y 2 z 1 , - + = 3 x 2 y 3 z 14 . - - + = (a) (i) Write the system of equations in matrix for
  26. November 2021, AS Core Pure, Q4 (5 marks): - . Her attempted 1 1 4 Anika thinks that, for two square matrices A and B, the inverse of AB is A - B proof of this is as follows. 1 1 1 1 ( AB A )( - B - ) A
  27. November 2021, AS Core Pure, Q6 (12 marks): J N 1 1 + m KK OO, where m is a non-zero 6 A transformation T of the plane has associated matrix M = 1 1 - - m constant. L P (a) (i) Show that T reverses orient
  28. November 2020, Core Pure, Q2 (6 marks): J N 0 1 a K O KK OO and 2 (a) The matrices M N 1 c OO are such that MN = I. = = - KK 1 b 0 L P 1 1 - L P Find a, b and c. [5] (b) State with a reason whether or
  29. November 2020, Core Pure, Q9 (8 marks): J N 1 2 - KK OO, where m is a 9 A linear transformation of the plane is represented by the matrix M = 3 m constant. L P (a) Find the set of values of m for whic
  30. November 2020, Core Pure, Q15 (17 marks): (a) Show that the three planes with equations x y z 3 12 + m + = - x y z 2 5 11 + + = - x y z 2 2 9 - + = - where m is a constant, meet at a unique point except
  31. November 2020, AS Core Pure, Q4 (4 marks): K O 4 The matrix M is 1 0 0 OO. KK 0 0 1 L P (a) (i) Calculate det M. [1] (ii) State two geometrical consequences of this value for the transformation associate
  32. November 2020, AS Core Pure, Q6 (8 marks): J N J N 2 1 m n KK OO and 1 KK OO respectively, where m and n are constants. 6 The matrices M and N are 2 m n L P L P (a) Investigate whether M and N are commut
  33. November 2020, AS Core Pure, Q8 (7 marks): J N 0 1 - KK OO. 8 (a) The matrix M is 1 0 - L P (i) Find M2. [1] (ii) Write down the transformation represented by M. [1] (iii) Hence state the geometrical sig
  34. November 2020, AS Core Pure, Q9 (7 marks): Three planes have equations kx y 2 z 0 + - = 2 x 3 y 6 z 5 + - = - 3 x 2 y 5 z 1 - + = where k is a constant. Investigate the arrangement of the planes for each
  35. June 2019, Core Pure, Q3 (7 marks): J N J N 3 1 k 1 KK OO and B KK OO, where k is a constant. 3 Matrices A and B are defined by A = = 2 1 2 0 L P L P 1 1 1 (a) Verify the result ( AB ) - B - A - i
  36. June 2019, Core Pure, Q11 (12 marks): (a) Specify fully the transformations represented by the following matrices. J N 3 4 - K O 5 5 • M1 = K O 3 4 5 5 L P J N 1 0 • M KK OO [4] 2 = 0 1 - L P (b) Fi
  37. June 2019, Core Pure, Q14 (13 marks): Three planes have equations x ay 2 - + = 2 x 3 y z 3 + + =- x by z c + + = where a, b and c are constants. (a) In the case where the planes do not intersect at
  38. June 2019, AS Core Pure, Q3 (6 marks): In this question you must show detailed reasoning. J N 2 1 1 = - KK OO. 1 A and B are matrices such that B - A - 1 1 L P (a) Find AB. [3] J N 1 1 KK OO, find B.
  39. June 2019, AS Core Pure, Q4 (8 marks): K O (a) Find M -1, where M 4 1 1 2 OO. [1] = - KK 2 1 2 - L P (b) Hence find, in terms of the constant k, the point of intersection of the planes x 2 y 3 z 19 ,
  40. June 2019, AS Core Pure, Q6 (11 marks): J N k m KK OO, and m 6 A linear transformation T of the x-y plane has an associated matrix M, where M = 1 k - m and k are real constants. L P (a) You are given
  41. June 2018, AS Core Pure, Q1 (4 marks): The matrices A, B and C are defined as follows: J N 1 K O c 2 0 3 m ^ h. A B C 2 , , 1 3 = KK OO = = 1 1 3 - 3 L P [4] Calculate all possible products formed fr
  42. June 2018, AS Core Pure, Q5 (7 marks): c cos 2 sin m, where i is a positive acute i i 5 A transformation of the x-y plane is represented by the matrix 2 sin cos i - i angle. (i) Write down the image
  43. June 2018, AS Core Pure, Q6 (4 marks): Find the invariant line of the transformation of the x-y plane represented by the matrix 2 c 0 m. 6 [4] 4 1 -
  44. June 2018, AS Core Pure, Q8 (6 marks): c m n c n m for all positive integers n. Prove by induction that 1 1 1 2 1 - 8 [6] = n 0 2 0 2
  45. June 2018, AS Core Pure, Q10 (8 marks): Three planes have equations -x + 2y + z = 0 2x - y - z = 0 x + y = a where a is a constant. (i) Investigate the arrangement of the planes: • when a 0 ; = • when

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 MEI A-Level Further Maths paper.

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