plumiFurther vectors: AQA A‑Level Further Maths questions
Every AQA A-Level Further Maths exam question on F Further vectors, 2019–2025, with its mark scheme and examiner’s report.
Paper
All papers
Exam
All years
All 31 questions on Further vectors, as a list
- June 2025, Paper 1, Q9 (4 marks): The vectors a and b are such that |a × b| = 12 a●b = −3 |a| = 7 Find the value of |b| [4 marks]
- June 2025, Paper 1, Q15 (9 marks): Three planes have equations 2 9 3 3 3 4 x y z x y z t x y z t + − = − + = + + = where t is a constant. The planes meet along a line of intersection. 15 (a) Find
- June 2025, Paper 2, Q1 (1 mark): The vectors a b and 1 c are perpendicular. Which one of the following statements must be true? Tick () one box. [1 mark] a = bc b = ac a = –b
- June 2025, Paper 2, Q11 (12 marks): The line L has vector equation = + − r 5 2 2 1 11 3 λ The point A has coordinates ( ) − − 1, 1, 6 11 (a) Find the coordi
- June 2025, Paper 3 Mechanics, Q7 (3 marks): The force F = 2 –4 3 acts at the point with coordinates (3, 5, –1) Show that the magnitude of the moment of this force about the origin is k√ 6 where k is a pos
- June 2025, AS Paper 1, Q10 (8 marks): The 3× 3 matrix M represents a reflection in the plane z = 0 The point A has position vector 2 5 4 The point A′ is the image of A under the reflection represent
- June 2024, Paper 1, Q5 (5 marks): The points A, B and C have coordinates A(5, 3, 4), B(8, –1, 9) and C(12, 5, 10) The points A, B and C lie in the plane ∏ 5 (a) Find a vector that is normal to t
- June 2024, Paper 1, Q12 (10 marks): The line L1 has equation r = 4 2 1 + λ 1 3 –1 The transformation T is represented by the matrix 2 1 0 3 4 6 –5 2 –3 The transformation T transforms the line L1
- June 2024, Paper 2, Q8 (4 marks): The vectors a, b, and c are such that a × b = 2 1 0 and a × c = 0 0 3 Work out (a – 4b + 3c) × (2a) [4 marks]
- June 2024, AS Paper 1, Q4 (1 mark): The line L has vector equation r = [ 4 ] –7 0 + λ[ –9 ] 1 3 Give the equation of L in Cartesian form. Tick () one box. [1 mark] x + 4 –9 = y – 7 1 = z 3 x – 4
- June 2024, AS Paper 1, Q5 (5 marks): The vectors a and b are given by a = 3i + 4j – 2k and b = 2i – j – 5k 5 (a) Calculate a.b [1 mark] �������������������������������������������������������������
- June 2023, Paper 1, Q9 (9 marks): The position vectors of the points A, B and C are a = 2i + j + 2k b = −i − 8j + 2k c = −2j respectively. 9 (a) Find the area of the triangle ABC [4 marks] 9 (b)
- June 2023, Paper 2, Q11 (9 marks): The line l1 passes through the points A(6, 2, 7) and B(4, −3, 7) 11 (a) Find a Cartesian equation of l1 [2 marks] 11 (b) The line l2 has vector equation r = 8 9
- June 2023, AS Paper 1, Q2 (1 mark): The two vectors a and b are such that a.b = 0 State the angle between the vectors a and b Circle your answer. [1 mark] 0° 45° 90° 180°
- June 2022, Paper 1, Q10 (12 marks): In this question all measurements are in centimetres. A small, thin laser pen is set up with one end at A(7, 2, −3) and the other end at B(9, −3, −2) A laser be
- June 2022, AS Paper 1, Q7 (9 marks): The lines l1 and l2 have equations l1 : r = 3 1 −2 2 664 3 775 + l 3 −4 1 2 664 3 775 l2 : r = −12 a −3 2 664 3 775 + m 3 2 −1 2 664 3 775 7 (a) Show that the p
- November 2021, Paper 1, Q11 (12 marks): The line L1 has equation r = 2 2 3 2 64 3 75 + l 2 3 −1 2 64 3 75 The line L2 has equation r = 6 4 1 2 64 3 75 + m −2 1 1 2 64 3 75 11 (a) Find the acute angle
- November 2021, Paper 1, Q12 (14 marks): The matrix A = 1 5 3 4 −2 p 8 5 −11 2 64 3 75 , where p is a constant. 12 (a) Given that A is a non-singular matrix, find A−1 in terms of p. State any restricti
- November 2021, Paper 2, Q3 (1 mark): The line L has equation r = 3 2 0 2 64 3 75 + l −1 −2 5 2 64 3 75 Which of the following lines is perpendicular to the line L? Tick (3) one box. [1 mark] r = 2
- November 2021, Paper 2, Q11 (9 marks): The Cartesian equation of the line L1 is x + 1 3 = −y + 5 2 = 2z + 5 3 The Cartesian equation of the line L2 is 2x − 1 2 = y − 14 m = z + 12 p The non-singular
- November 2021, AS Paper 1, Q5 (2 marks): Show that the vectors 1 −3 5 2 64 3 75 and 7 4 1 2 64 3 75 are perpendicular. [2 marks]
- November 2021, AS Paper 1, Q15 (8 marks): Two submarines are travelling on different straight lines. The two lines are described by the equations r = 2 −1 4 2 64 3 75 + l 5 3 −2 2 64 3 75 and x − 5 4 =
- November 2020, Paper 1, Q11 (11 marks): The lines l1, l2 and l3 are defined as follows. l1 : r − 1 5 −1 2 664 3 775 0 B B @ 1 C C A × −2 1 −3 2 664 3 775 = 0 l2 : r − −3 2 7 2 664 3 775 0 B B @ 1 C C
- November 2020, Paper 2, Q1 (1 mark): Three of the four expressions below are equivalent to each other. Which of the four expressions is not equivalent to any of the others? Circle your answer. [1 m
- November 2020, Paper 2, Q15 (16 marks): The points A(7, 2, 8), B(7, ×4, 0) and C(3, 3.2, 9.6) all lie in the plane P. 15 (a) Find a Cartesian equation of the plane P. [3 marks] 15 (b) The line L1 has
- November 2020, AS Paper 1, Q13 (9 marks): Line l1 has equation x − 2 3 = 1 − 2y 4 = −z and line l2 has equation r = −7 4 −2 2 64 3 75 + m 12 a + 3 2b 2 64 3 75 13 (a) In the case when l1 and l2 are para
- June 2019, Paper 1, Q5 (3 marks): A plane has equation r. 1 1 1 2 64 3 75 = 7 A line has equation r = 2 0 1 2 64 3 75 + m 1 0 1 2 64 3 75 Calculate the acute angle between the line and the plane
- June 2019, Paper 1, Q10 (8 marks): The points A(5, −4, 6) and B(6, −6, 8) lie on the line L. The point C is (15, −5, 9). 10 (a) D is the point on L that is closest to C. Find the coordinates of D
- June 2019, Paper 2, Q7 (6 marks): The points A, B and C have coordinates A(4, 5, 2), B(−3, 2, −4) and C(2, 6, 1) 7 (a) Use a vector product to show that the area of triangle ABC is 5 11 p 2 [4 m
- June 2019, Paper 2, Q11 (8 marks): The line L1 has equation x − 2 3 = y + 4 8 = 4z − 5 5 The line L2 has equation r − −2 0 3 2 64 3 75 0 B @ 1 C A ≥ 2 1 3 2 64 3 75 = 0 Find the shortest distance
- June 2019, AS Paper 1, Q13 (10 marks): Line l1 has Cartesian equation x − 3 = 2y + 2 3 = 2 − z 13 (a) Write the equation of line l1 in the form r = a + lb where l is a parameter and a and b are vecto
Questions, mark schemes and examiner’s reports are © AQA, from their published past papers. Plumi isn’t affiliated with any exam board. Want whole papers? Every AQA A-Level Further Maths paper.
Rather write your answers by hand on iPad? Plumi marks them against the same mark schemes.