Two lines have equations r = i + 2j + k + λ(ai + 2j – k) and r = 2i + j – k + µ(2i – j + k), where a is a constant.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2020). The mark or mark type is shown in brackets.

1. [B1] Express general point of at least one line correctly in component form, i.e. (1 + aλ, 2 + 2λ, 1 – λ) or (2 + 2µ, 1 – µ, –1 + µ)
2. [M1] Equate at least two pairs of corresponding components and solve for λ or for µ
3. [A1] Obtain λ = –3 or µ = 5
4. [A1] Obtain a = -11/3
5. [A1] State that the point of intersection has position vector 12i - 4j + 4k
6. [M1] Use correct process for finding the scalar product of direction vectors for the two lines
7. [*M1] Using the correct process for the moduli, divide the scalar product by the product of the moduli and equate the result to ±1/6
8. [A1] State a correct equation in a in any form, e.g. (2a-3) / (sqrt(6) * sqrt(a²+5)) = ±1/6
9. [DM1] Solve 3-term quadratic for a having expanded (2a – 3)² to produce 3 terms e.g. 36(2a – 3)² = 6(a² + 5) 138a² – 432a + 294 = 0 23a² – 72a + 49 = 0 (23a – 49)(a – 1) = 0
10. [A1] Obtain a = 1
11. [A1] Obtain a = 49/23
12. [M1] Use of cosine rule. Must be correct vectors.