Angles with the scalar product: OCR MEI A‑Level Further Maths questions

OCR MEI 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.

All 18 questions on Angles with the scalar product, as a list

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] - + = + + =
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] +
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
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
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
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] + - + + =
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
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 -
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
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
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
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
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
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
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]
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
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 ,
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

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