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Further vectors: OCR MEI A‑Level Further Maths questions

Every OCR MEI A-Level Further Maths exam question on CP.6 Further vectors, 2018–2025, with its mark scheme and examiner’s report.

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847 questions

  1. Q2

    June 2025 · Core Pure

    4 marksCP.6 Further vectors
    The whole paper ↗
    June 2025 Core Pure, question 2: In this question you must show detailed reasoning. Find the acute angle between the planes x 2 y 2 z 5 and x 2 y z 8 . [4] - + = + + =

  2. Q11

    June 2025 · Core Pure

    14 marksCP.6 Further vectors
    The whole paper ↗
    June 2025 Core Pure, question 11: The lines l1 and l2 have equations y 1 x z 3 + 2 - - l1: l2: r 2 i j 3 k ( i cj 2 k ) = = = + + + m - + + a b 1 where a, b and c are constants. (a) In the case

  3. Q2

    June 2025 · AS Core Pure

    4 marksCP.6 Further vectors
    The whole paper ↗
    June 2025 AS Core Pure, question 2: In this question you must show detailed reasoning. Find the acute angle between the vector i 3 2 j - and the normal vector to the plane k + 2 x 3 y z 6 . [4] +

  4. Q5

    June 2024 · Core Pure

    6 marksCP.6 Further vectors
    The whole paper ↗
    June 2024 Core Pure, question 5: K O K O K O 5 (a) Given that u = 1 OO, v = 0 OO and u v b OO, find a and b. [3] KK KK = KK # 2 1 3 L P L P L P (b) Using u # , determine the angle between the v

  5. Q11

    June 2024 · Core Pure

    14 marksCP.6 Further vectors
    The whole paper ↗
    June 2024 Core Pure, question 11: The plane P has equation x 2 y 2 z 4 . The point P has coordinates (8, 4, 5). - + = (a) Calculate the shortest distance from P to P. [2] The line L has equation

  6. Q9

    June 2024 · AS Core Pure

    8 marksCP.6 Further vectors
    The whole paper ↗
    June 2024 AS Core Pure, question 9: In this question you must show detailed reasoning. Find a vector v which has the following properties. • It is a unit vector. • It is parallel to the plane x y

  7. Q2

    June 2023 · Core Pure

    5 marksCP.6 Further vectors
    The whole paper ↗
    June 2023 Core Pure, question 2: In this question you must show detailed reasoning. Find the angle between the vector i 3 2 j + and the plane k x 3 y 2 z 8 . [5] + - + + =

  8. Q16

    June 2023 · Core Pure

    10 marksCP.6 Further vectors
    The whole paper ↗
    June 2023 Core Pure, question 16: and the line x 3 y 1 z + , 5 - - 16 The point P (4, 1, 0) is equidistant from the plane x 2 y 2 z 0 + + = = = 2 b 3 where b 2 0 . Determine the value of b. [10]

  9. Q3

    June 2023 · Extra Pure

    8 marksFP.10 Multivariable calculusCP.6 Further vectors
    The whole paper ↗
    June 2023 Extra Pure, question 3: 2 2 3 A surface, S, is defined by ( , , ) g x y z 0 where ( , , ) g x y z 2 x x y 2 xy 27 z . The normal to = = - + + a k and the tangent plane to S at the poin

  10. Q10

    June 2023 · AS Core Pure

    7 marksCP.6 Further vectors
    The whole paper ↗
    June 2023 AS Core Pure, question 10: The plane P has normal vector 2i + aj - k, where a is a positive constant, and the point (3, -1, 1) lies in P. The plane x - = makes an angle of 45° with P. z 3

  11. Q13

    June 2022 · Core Pure

    17 marksCP.6 Further vectors
    The whole paper ↗
    June 2022 Core Pure, question 13: The points A and B have coordinates (4, 0, -1) and (10, 4, -3) respectively. The planes P and 1 P have equations x y = and x y z 2 5 2 3 - = - respectively. 4 -

  12. Q2

    June 2022 · AS Core Pure

    7 marksCP.6 Further vectors
    The whole paper ↗
    June 2022 AS Core Pure, question 2: (a) Show that the vector i j k x y z [3] 4 2 is parallel to the plane 2 3 10 . + + + - = (b) Determine the acute angle between the planes x y z x y z [4] 2 3 10

  13. Q11

    November 2021 · Core Pure

    9 marksCP.6 Further vectors
    The whole paper ↗
    November 2021 Core Pure, question 11: (a) Given that u = mi + j - 3k and v = i + 2j - 2k, find the following, giving your answers in terms of m. (i) u.v [1] (ii) u #v [2] (b) Hence determine (i) the

  14. Q5

    November 2021 · Extra Pure

    6 marksFP.10 Multivariable calculusCP.6 Further vectors
    The whole paper ↗
    November 2021 Extra Pure, question 5: 2 2 5 A surface S is defined for z 0 by x y 2 z 126 . C is the set of points on S for which the H + + = 1 r radians. tangent plane to S at that point intersects

  15. Q3

    November 2021 · AS Core Pure

    7 marksCP.3 MatricesCP.6 Further vectors
    The whole paper ↗
    November 2021 AS Core Pure, question 3: 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

  16. Q8

    November 2020 · Core Pure

    9 marksCP.6 Further vectors
    The whole paper ↗
    November 2020 Core Pure, question 8: K O K O K O K O 8 (a) Given that the lines r 2 1 OO and r 2 3 OO meet, determine k. [5] = KK OO + m KK = KK OO + n KK 2 3 k 4 L P L P L P L P (b) In this questi

  17. Q15

    November 2020 · Core Pure

    17 marksCP.3 MatricesCP.6 Further vectors
    The whole paper ↗
    November 2020 Core Pure, question 15: (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

  18. Q6

    November 2020 · Extra Pure

    17 marksFP.10 Multivariable calculusCP.6 Further vectors
    The whole paper ↗
    November 2020 Extra Pure, question 6: 4 2 2 6 A surface S is defined by z f ( , ) x y 4 x 4 y 17 x y . = = + - (a) (i) Show that there is only one stationary point on S. [5] The value of z at the st

  19. Q10

    November 2020 · AS Core Pure

    7 marksCP.6 Further vectors
    The whole paper ↗
    November 2020 AS Core Pure, question 10: A vector v has magnitude 1 unit. The angle between v and the positive z-axis is 60°, and v is parallel to the plane x 2 y 0 . - = Given that v = a i b j c k , w

  20. Q2

    June 2019 · Core Pure

    3 marksCP.6 Further vectors
    The whole paper ↗
    June 2019 Core Pure, question 2: The plane x 2 y cz 4 is perpendicular to the plane x 2 cy 6 z 9 , where c is a constant. + + = - + = Find the value of c. [3]
All 24 questions on Further vectors, as a list
  1. June 2025, Core Pure, Q2 (4 marks): In this question you must show detailed reasoning. Find the acute angle between the planes x 2 y 2 z 5 and x 2 y z 8 . [4] - + = + + =
  2. June 2025, Core Pure, Q11 (14 marks): The lines l1 and l2 have equations y 1 x z 3 + 2 - - l1: l2: r 2 i j 3 k ( i cj 2 k ) = = = + + + m - + + a b 1 where a, b and c are constants. (a) In the case
  3. June 2025, AS Core Pure, Q2 (4 marks): In this question you must show detailed reasoning. Find the acute angle between the vector i 3 2 j - and the normal vector to the plane k + 2 x 3 y z 6 . [4] +
  4. June 2024, Core Pure, Q5 (6 marks): K O K O K O 5 (a) Given that u = 1 OO, v = 0 OO and u v b OO, find a and b. [3] KK KK = KK # 2 1 3 L P L P L P (b) Using u # , determine the angle between the v
  5. June 2024, Core Pure, Q11 (14 marks): The plane P has equation x 2 y 2 z 4 . The point P has coordinates (8, 4, 5). - + = (a) Calculate the shortest distance from P to P. [2] The line L has equation
  6. June 2024, AS Core Pure, Q9 (8 marks): In this question you must show detailed reasoning. Find a vector v which has the following properties. • It is a unit vector. • It is parallel to the plane x y
  7. June 2023, Core Pure, Q2 (5 marks): In this question you must show detailed reasoning. Find the angle between the vector i 3 2 j + and the plane k x 3 y 2 z 8 . [5] + - + + =
  8. June 2023, Core Pure, Q16 (10 marks): and the line x 3 y 1 z + , 5 - - 16 The point P (4, 1, 0) is equidistant from the plane x 2 y 2 z 0 + + = = = 2 b 3 where b 2 0 . Determine the value of b. [10]
  9. June 2023, Extra Pure, Q3 (8 marks): 2 2 3 A surface, S, is defined by ( , , ) g x y z 0 where ( , , ) g x y z 2 x x y 2 xy 27 z . The normal to = = - + + a k and the tangent plane to S at the poin
  10. June 2023, AS Core Pure, Q10 (7 marks): The plane P has normal vector 2i + aj - k, where a is a positive constant, and the point (3, -1, 1) lies in P. The plane x - = makes an angle of 45° with P. z 3
  11. June 2022, Core Pure, Q13 (17 marks): The points A and B have coordinates (4, 0, -1) and (10, 4, -3) respectively. The planes P and 1 P have equations x y = and x y z 2 5 2 3 - = - respectively. 4 -
  12. June 2022, AS Core Pure, Q2 (7 marks): (a) Show that the vector i j k x y z [3] 4 2 is parallel to the plane 2 3 10 . + + + - = (b) Determine the acute angle between the planes x y z x y z [4] 2 3 10
  13. November 2021, Core Pure, Q11 (9 marks): (a) Given that u = mi + j - 3k and v = i + 2j - 2k, find the following, giving your answers in terms of m. (i) u.v [1] (ii) u #v [2] (b) Hence determine (i) the
  14. November 2021, Extra Pure, Q5 (6 marks): 2 2 5 A surface S is defined for z 0 by x y 2 z 126 . C is the set of points on S for which the H + + = 1 r radians. tangent plane to S at that point intersects
  15. 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
  16. November 2020, Core Pure, Q8 (9 marks): K O K O K O K O 8 (a) Given that the lines r 2 1 OO and r 2 3 OO meet, determine k. [5] = KK OO + m KK = KK OO + n KK 2 3 k 4 L P L P L P L P (b) In this questi
  17. 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
  18. November 2020, Extra Pure, Q6 (17 marks): 4 2 2 6 A surface S is defined by z f ( , ) x y 4 x 4 y 17 x y . = = + - (a) (i) Show that there is only one stationary point on S. [5] The value of z at the st
  19. November 2020, AS Core Pure, Q10 (7 marks): A vector v has magnitude 1 unit. The angle between v and the positive z-axis is 60°, and v is parallel to the plane x 2 y 0 . - = Given that v = a i b j c k , w
  20. June 2019, Core Pure, Q2 (3 marks): The plane x 2 y cz 4 is perpendicular to the plane x 2 cy 6 z 9 , where c is a constant. + + = - + = Find the value of c. [3]
  21. June 2019, Core Pure, Q12 (9 marks): Three intersecting lines L1, L2 and L3 have equations x y z x y z x y z 1 2 4 - - + L : , L : and L : . = = = = - = = 2 3 1 1 2 4 1 1 5 1 2 3 Find the area of t
  22. 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 ,
  23. June 2018, AS Core Pure, Q2 (3 marks): K O K O 2 [3] Find, to the nearest degree, the angle between the vectors 0 OO and 3 OO. KK KK 2 3 - - L P L P
  24. 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

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