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A-LevelFurther MathematicsVectors9231/12 · May/June 2025 · Q616 Marks

The points A, B, C have position vectors i-2k, i+2j+2k, 2i-j-k, respectively.

  1. (a)Find the equation of the plane ABC, giving your answer in the form ax+by+cz = d.[5]
  2. (b)A point D has position vector i+tk, where t≠-2. Find the acute angle between the planes ABC and ABD.[4]
  3. (c)Find the values of t such that the shortest distance between the lines AB and CD is √2.[7]

Mark scheme

Marking points from the official mark scheme (May/June 2025). The mark or mark type is shown in brackets.

  1. [B1] Finds direction vectors of two lines in the plane.
  2. [M1 A1] Finds normal to the plane ABC.
  3. [M1 A1] Substitutes point.
  4. [M1 A1] Finds normal to the plane ABD.
  5. [M1] Uses dot product of normal vectors in correct formula.
  6. [A1] 0.641 rad.
  7. [M1 A1] Finds common perpendicular.
  8. [M1 A1] Uses formula for perpendicular distance.
  9. [M1] Sets equal to √2 and solves for t.
  10. [A1 A1] t = 6.83, t = 1.17.

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About this question

Topic
Vectors, A-Level Further Mathematics (syllabus 9231). Structured question, 16 marks.
Source
A-Level Further Mathematics, May/June 2025 Paper 1 Variant 2, question 6.

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