plumiFurther vectors: OCR A‑Level Further Maths questions
Every OCR A-Level Further Maths exam question on CP.6 Further vectors, 2018–2025, with its mark scheme and examiner’s report.
Find questions648
![June 2024 AS Pure Core, question 3: K O K O 3 (a) (i) Find 2 5 OO. [1] KK OO KK # 1 2 - - L P L P (ii) State a geometrical relationship between the answer to part (a)(i) and the J N J N 1 3 K O K](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/y53101-jun24-03-qp.webp?v=3)
![June 2024 AS Pure Core, question 5: The line through points A 8 ( , 7 , 2 ) and B 11 ( , 9 0 , ) is denoted by L1. - - - (a) Find a vector equation for L1. [2] (b) Determine whether the point ( 26](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/y53101-jun24-05-qp.webp?v=3)
![June 2022 Pure Core 1, question 4: K O K O 4 Determine the acute angle between the line r 1 2 3 OO and the y-axis. [4] = + KK OO m KK 3 3 - L P L P](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/y54001-jun22-04-qp.webp?v=3)
![June 2022 Pure Core 2, question 1: (a) Find a vector which is perpendicular to both 3i - 5j - k and i + 3j - 4k. [1] The equations of two lines are r = 2i + 3j + 3k+ m(i - 2j + k) and r = i + 11j](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/y54101-jun22-01-qp.webp?v=3)
All 36 questions on Further vectors, as a list
- June 2025, Pure Core 1, Q5 (8 marks): K O K O K O 5 A vector equation of the plane P is r 4 1 1 OO. = + + KK OO m KK OO n KK 1 3 1 0 - L P L P L P (a) Verify that a cartesian equation of P is x y 2
- June 2025, Pure Core 1, Q7 (8 marks): A 3-D coordinate system, whose units are metres, is set up to model a street containing telephone cables T1 and T2. The cables are modelled as straight lines wi
- June 2025, Pure Core 2, Q1 (3 marks): In this question you must show detailed reasoning. J N J N 2 1 - K O K O Vectors a and b are given by a 4 OO and b 3 OO. = - KK = KK 3 2 - L P L P Determine, in
- June 2025, AS Pure Core, Q2 (5 marks): K O K O 2 Two vectors, a and b, are given by a O where p is a constant. 3 OO and b 6 K O = - KK = K 13 p L P L P (a) Find expressions in terms of p for each of
- June 2025, AS Pure Core, Q6 (7 marks): K O K O K O K O 6 The equations of two lines, l1 and l2, are l r 1 27 OO and l r 10 10 OO. = - KK OO + m KK - = KK OO + n KK 1 | 2 | 3 19 10 10 - - L P L P L P
- June 2025, AS Additional Pure Mathematics, Q2 (6 marks): K O K O K O 2 The vectors p, q and r are such that p 3 , q 4 OO and r 1 OO. = KK OO = - KK = KK 7 9 2 - L P L P L P (a) (i) Determine the value of the integer k
- June 2024, Pure Core 1, Q11 (7 marks): A 3-D coordinate system, whose units are metres, is set up to model a construction site. The construction site contains four vertical poles P1, P2, P3 and P4. T
- June 2024, Pure Core 2, Q5 (6 marks): Vectors, a, b and c, are given by a i ( 1 p ) j ( p 2 ) , k b 2 i j k and = + - + + = + + c i 14 j ( p 3 ) k where p is a constant. = + + - You are given that a
- June 2024, Additional Pure Mathematics, Q4 (10 marks): K O K O 4 The vectors a and b are given by a q 2 OO and b 2 q 5 OO, where p, q and r are real numbers. = KK + = KK - 2 r 3 r 3 - + L P L P (a) Given that b is n
- June 2024, AS Pure Core, Q3 (7 marks): K O K O 3 (a) (i) Find 2 5 OO. [1] KK OO KK # 1 2 - - L P L P (ii) State a geometrical relationship between the answer to part (a)(i) and the J N J N 1 3 K O K
- June 2024, AS Pure Core, Q5 (10 marks): The line through points A 8 ( , 7 , 2 ) and B 11 ( , 9 0 , ) is denoted by L1. - - - (a) Find a vector equation for L1. [2] (b) Determine whether the point ( 26
- June 2023, Pure Core 1, Q8 (15 marks): The points P, Q and R have coordinates (0, 2, 3), (2, 0, 1) and (1, 3, 0) respectively. The acute angle between the line segments PQ and PR is i. 2 (a) Show tha
- June 2023, Pure Core 2, Q6 (8 marks): K O K O K O 6 The equation of the plane % is r 2 4 3 OO. = + + KK OO m KK OO n KK 1 3 1 L P L P L P J N 2 K O (a) Find the acute angle between % and the plane w
- June 2023, Additional Pure Mathematics, Q3 (7 marks): The points A and B have position vectors a i p j q k and b 2 i 3 j 2 k respectively, = + + = + + relative to the origin O. (a) Determine the value of p and the
- June 2023, AS Pure Core, Q2 (8 marks): The lines L1 and L2 have the following equations. J N J N 5 5 - K O K O L : r 6 2 = + - KK OO m KK OO 1 15 2 - L P L P J N J N 24 3 K O K O L : r 1 1 = + KK OO
- June 2023, AS Pure Core, Q4 (4 marks): K O 4 The vector p, all of whose components are positive, is given by p a 5 OO where a is a constant. = KK - 26 L P J N 2 K O You are given that p is perpendicu
- June 2023, AS Additional Pure Mathematics, Q4 (7 marks): The equation of line l can be written in either of the following vector forms. • r a b where , = + m m ! R • ( r c ) d 0 - = # (a) Write down two equations invo
- June 2022, Pure Core 1, Q4 (4 marks): K O K O 4 Determine the acute angle between the line r 1 2 3 OO and the y-axis. [4] = + KK OO m KK 3 3 - L P L P
- June 2022, Pure Core 2, Q1 (6 marks): (a) Find a vector which is perpendicular to both 3i - 5j - k and i + 3j - 4k. [1] The equations of two lines are r = 2i + 3j + 3k+ m(i - 2j + k) and r = i + 11j
- June 2022, Pure Core 2, Q10 (8 marks): The coordinates of the points A and B are (3, -2, -1) and (13, 10, 9) respectively. • The plane %A contains A and the plane %B contains B. • The planes %A and %
- June 2022, AS Pure Core, Q1 (8 marks): K O K O 1 (a) Determine whether the point (19, -12, 17) lies on the line r 2 2 OO. [3] = - KK OO + m KK - 7 4 L P L P J N J N 1 3 - K O K O Vectors a and b are
- June 2022, AS Pure Core, Q8 (9 marks): The line segment AB is a diameter of a sphere, S. The point C is any point on the surface of S. (a) Explain why AC BC 0 for all possible positions of C. [3] = :
- November 2021, Pure Core 1, Q4 (11 marks): Points A, B and C have coordinates (4, 2, 0), (1, 5, 3) and (1, 4, -2) respectively. The line l passes through A and B. (a) Find a cartesian equation for l. [3]
- November 2021, Pure Core 2, Q3 (9 marks): K O K O 3 The line l1 has equation r 3 2 OO. = - KK OO + m KK 3 2 - L P L P J N 2 K O The plane P has equation r. 5 4 . KK - OO = 3 - L P (a) Find the position
- November 2021, AS Pure Core, Q1 (5 marks): The lines l1 and l2 have the following equations. J N J N 8 2 - K O K O l r 11 5 | = - + KK OO m KK OO 1 2 3 - L P L P J N J N 6 3 - - K O K O l r 11 1 | = + KK
- November 2021, AS Pure Core, Q9 (13 marks): The points P(3, 5, 21 ) and Q( 1 - , 3, 16 ) are on the ceiling of a long straight underground tunnel. - - A ventilation shaft must be dug from the point M on t
- November 2020, Pure Core 1, Q6 (5 marks): The equations of two non-intersecting lines, l1 and , l2 are J N J N J N J N 1 2 2 1 K O K O K O K O l : r 2 1 OO , l : r 2 1 . = KK OO + m KK = KK OO + n KK -
- November 2020, Pure Core 2, Q4 (9 marks): The equations of two intersecting lines l1 and l2 are J N J N J N J N 1 2 7 1 - K O K O K O K O l : r 0 1 l : r 9 1 = KK OO + m KK OO = KK OO + n KK OO 1 2 a 3
- November 2020, Additional Pure Mathematics, Q4 (7 marks): Points A, B and C have position vectors a, b and c respectively, relative to origin O. It is given that b c a and that a = . 3 = # (a) Determine each of the fol
- November 2020, AS Pure Core, Q7 (6 marks): The equations of two intersecting lines are J N J N J N J N 12 2 2 3 - - K O K O K O K O r = a 2 OO r = 0 1 + + KK OO m KK KK OO n KK OO 1 1 5 1 - - L P L P L P
- June 2019, Pure Core 1, Q8 (6 marks): The equation of a plane is x 4 2 y z 7 . + + = The point A has coordinates (9, 6, 1) and the point B is the reflection of A in the plane. Find the coordinates o
- June 2019, Pure Core 2, Q2 (8 marks): K O 2 (a) A plane P has the equation r. 6 15 . C is the point (4, -5, 1). KK OO = 2 - L P Find the shortest distance between P and C. [3] (b) Lines l1 and l2 ha
- June 2019, AS Pure Core, Q3 (10 marks): The position vector of point A is a 9 i 2 j 6 k . =- + + The line l passes through A and is perpendicular to a. (a) Determine the shortest distance between the
- June 2019, AS Additional Pure Mathematics, Q3 (4 marks): The non-zero vectors x and y are such that x # y = 0. (a) Explain the geometrical significance of this statement. [2] (b) Use your answer to part (a) to explain
- June 2019, AS Additional Pure Mathematics, Q8 (13 marks): The motion of two remote controlled helicopters P and Q is modelled as two points moving along straight lines. J N J N p q 2 4 5 8 + + K O K O Helicopter P move
- June 2018, AS Pure Core, Q1 (5 marks): f p and f p. 1 (i) Find a vector which is perpendicular to both [2] 3 6 - 2 4 - (ii) The cartesian equation of a line is x y z 3 2 + . 4 = - = 2 [3] Express the
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.