The points A and B have position vectors i + 2j – k and 3i + j + k respectively. The line l has equation r = 2i + j + k + μ(i + j + 2k).

Mark scheme

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

1. [M1] Carry out correct method for finding a vector equation for AB
2. [A1] Obtain (r =)i+2j-k+λ(2i-j+2k), or equivalent
3. [M1] Equate two pairs of components of general points on their AB and l and solve for λ or for μ (1+2λ = 2+μ; 2-λ = 1+μ; -1+2λ = 1+2μ)
4. [A1] Obtain correct answer for λ or μ, e.g. λ = 0, μ = − 1
5. [A1] Verify that all three equations are not satisfied and the lines fail to intersect (≠ is sufficient justification e.g. 2 ≠ 0). Conclusion needs to follow correct values (Alternatives table shows different pairings of (λ,μ) for different components)
6. [B1] State or imply midpoint has position vector 2i + (3/2)j (2i + 1.5j + 0k is implied from 2i+ (3/2)j )
7. [M1] Substitute in 2x – y + 2z = d and find d (Correct use of their direction for AB and their midpoint)
8. [A1] Obtain plane equation 4x – 2y + 4z = 5 (or equivalent e.g. r . (2, -1, 2) = 2.5 (2, -1, 2), r . (4, -2, 4) = 5)
9. [M1] Substitute components of l in plane equation and solve for μ (Correct use of their plane equation.)
10. [A1] Obtain μ = -1/2 and position vector (3/2)i + (3/2)j for the point P (Final answer. Accept coordinates in place of position vector)