The line l₁ is parallel to the vector ai – j + k, where a is a constant, and passes through the point whose position vector is 9j + 2k. The line l₂ is parallel to the vector –ai + 2j + 4k and passes through the point whose position vector is –6i – 5j + 10k.

Mark scheme

Marking points from the official mark scheme (May/June 2018). The mark or mark type is shown in brackets.

1. B1 (i)(a) Point on l₁ is (0, 9, 2)ᵀ and on l₂ is (-6, -5, 10)ᵀ. Direction vectors (a, -1, 1)ᵀ and (-a, 2, 4)ᵀ.
2. M1 (i)(a) Equates coordinates of points. 0+aλ = -6-aμ, 9-λ = -5+2μ, 2+λ = 10+4μ
3. A1 (i)(a) Shows a = -6/13. (AG)
4. M1 A1 (i)(b) Uses cross product to find normal to the plane AEF. n = d₁ × d₂. Result is proportional to (-13i + 5j - k).
5. M1 (i)(b) Substitutes a point. e.g. 5(9)-2=43
6. A1 (i)(b) Equation of plane: -13x + 5y - z = 43
7. B1 (ii) Finds |d₁ × d₂| = √(36 + 26a²).
8. M1 (ii) Takes dot product of position vector connecting points (e.g., P₂-P₁) and d₁ × d₂. Result is 36 + 78a.
9. A1 (ii) Correct dot product, 36 + 78a.
10. M1 (ii) Puts distance equal to 3√(30). 3√(30) = |36 + 78a| / √(36 + 26a²).
11. A1 (ii) Solves for a. (a - 3)² = 0 ⇒ a = 3.