plumiAngles with the scalar product: OCR A‑Level Further Maths questions
OCR A-Level Further Maths past-paper questions on 6.3 Angles with the scalar product, 2018–2025, with mark schemes and examiner’s reports where available. Add MadAsMaths questions to your practice, or browse them separately by topic and difficulty.
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All 14 questions on Angles with the scalar product, as a list
- 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 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, 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 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, 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 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, 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 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, 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
- 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
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