Relative to the origin O, the point A has position vector given by OA = i + 2j + 4k. The line l has equation r = 9i – j + 8k + μ(3i – j + 2k).

Mark scheme

Marking points from the official mark scheme (May/June 2017). The mark or mark type is shown in brackets.

1. [B1] Find AP for a general point P on l with parameter λ, e.g.(8 + 3λ, – 3 – λ, 4 + 2λ)
2. [M1] Equate scalar product of AP and direction vector of l to zero and solve for λ
3. [A1] Obtain λ = -5/2 and foot of perpendicular (3/2)i + (3/2)j + 3k
4. [M1] Carry out a complete method for finding the position vector of the reflection of A in l
5. [A1] Obtain answer 2i + j + 2k
6. [B1] Find AP for a general point P on l with parameter λ, e.g.(8 + 3λ, – 3 – λ, 4 + 2λ)
7. [M1] Differentiate AP and solve for λ at minimum
8. [A1] Obtain λ = -5/2 and foot of perpendicular (3/2)i + (3/2)j + 3k
9. [M1] Carry out a complete method for finding the position vector of the reflection of A in l
10. [A1] Obtain answer 2i + j + 2k
11. [B1] Use scalar product to obtain an equation in a, b and c, e.g. 3a – b + 2c = 0
12. [M1] Form a second relevant equation, e.g. 9a – b + 8c = 0 and solve for one ratio, e.g. a : b
13. [A1] Obtain final answer a : b : c = 1 : 1 : − 1 and state plane equation x + y − z = 0
14. [M1] Attempt to calculate vector product of two relevant vectors, e.g. (3i-j+2k)×(9i-j+8k)
15. [A1] Obtain two correct components
16. [A1] Obtain correct answer, e.g. -6i-6j+6k, and state plane equation -x-y+z = 0
17. [M1] Using a relevant point and relevant vectors, attempt to form a 2-parameter equation for the plane, e.g. r = 6i + 6k + s(3i – j + 2k) + t(9i-j+8k)
18. [A1] State 3 correct equations in x, y, z, s and t
19. [A1] Eliminate s and t and state plane equation x+y-z = 0 , or equivalent
20. [M1] Using a relevant point and relevant vectors, attempt to form a determinant equation for the plane, e.g. (x-3) (y-1) (z-4) / (3 -1 2) / (9 -1 8) = 0
21. [A1] Expand a correct determinant and obtain two correct cofactors
22. [A1] Obtain answer -6x-6y+6z = 0, or equivalent
23. [M1] Using the correct processes, divide the scalar product of OA and a normal to the plane by the modulus of the normal or make a recognisable attempt to apply the perpendicular formula
24. [A1FT] Obtain a correct expression in any form, e.g. (1+2-4) / √(1²+1²+(-1)²) , or equivalent
25. [A1] Obtain answer 1/√3, or exact equivalent
26. [B1FT] Obtain equation of the parallel plane through A, e.g. x + y − z = -1
27. [M1] Use correct method to find its distance from the origin
28. [A1] Obtain answer 1/√3, or exact equivalent
29. [B1FT] Form equation for the intersection of the perpendicular through A and the plane
30. [M1] Solve for λ
31. [A1] |λn| = 1/√3