Two lines l and m have equations r = ai + 2j + 3k + λ(i – 2j + 3k) and r = 2i + j + 2k + μ(2i – j + k) respectively, where a is a constant. It is given that the lines intersect. (i) Find the value of a. (ii) When a has this value, find the equation of the plane containing l and m.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2019). The mark or mark type is shown in brackets.

1. [B1] Express general point of l or m in component form e.g. (a + λ, 2 – 2λ, 3 + 3λ) or (2 + 2μ, 1 – μ, 2 + μ)
2. [M1] Equate at least two pairs of corresponding components and solve for λ or for μ
3. [A1] Obtain either λ = − 2 or μ = -5 OR λ = (-a)/3 or μ = (a-1)/3 OR λ = (1/5)(a-4) or μ = (1/5)(3a-7)
4. [A1] Obtain a = - 6
5. [B1] Use scalar product to obtain a relevant equation in a, b and c, e.g. a - 2b + 3c = 0
6. [M1] Obtain a second equation, e.g. 2a - b + c = 0 and solve for one ratio
7. [A1] Obtain a:b:c = 1:5:3 OE
8. [M1] Substitute a relevant point and values of a, b, c in general equation and find d
9. [A1] Obtain correct answer x + 5y + 3z = 13 FT OE. The FT is on a from part (i), if used
10. [M1] Attempt to calculate vector product of relevant vectors, e.g. (i-2j+3k) x (2i-j+k)
11. [A1] Obtain two correct components
12. [A1] Obtain correct answer, e.g. i + 5j + 3k
13. [M1] Substitute a relevant point and find d
14. [A1] Obtain correct answer x + 5y + 3z = 13 FT OE. The FT is on a from part (i), if used