The point P has position vector 3i – 2j + k. The line l has equation r = 4i + 2j + 5k + μ(i + 2j + 3k).

Mark scheme

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

1. [B1] EITHER: Find PQ (or QP) for a general point Q on l, e.g. (1+μ) i+(4+2μ)j+(4+3μ)k
2. [M1] Calculate the scalar product of PQ and a direction vector for l and equate to zero
3. [A1] Solve and obtain correct solution e.g. μ = -3/2
4. [M1] Carry out method to calculate PQ
5. [A1] Obtain answer 1.22
6. [B1] OR1: Find PQ (or QP) for a general point Q on l
7. [M1] Use a correct method to express PQ² (or PQ) in terms of μ
8. [A1] Obtain a correct expression in any form
9. [M1] Carry out a complete method for finding its minimum
10. [A1] Obtain answer 1.22
11. [B1] OR2: Calling (4, 2, 5) A, state PA (or AP) in component form, e.g. i + 4j + 4k
12. [M1] Use a scalar product to find the projection of PA (or AP) on l
13. [A1] Obtain correct answer 21/√14 , or equivalent
14. [M1] Use Pythagoras to find the perpendicular
15. [A1] Obtain answer 1.22
16. [B1] OR3: State PA (or AP) in component form
17. [M1] Calculate vector product of PA and a direction vector for l
18. [A1] Obtain correct answer, e.g. 4i + j – 2k
19. [M1] Divide modulus of the product by that of the direction vector
20. [A1] Obtain answer 1.22
21. [B1] EITHER: Use scalar product to obtain a relevant equation in a, b and c, e.g. a + 2b + 3c = 0
22. [M1] Obtain a second relevant equation, e.g. using PA • a + 4b+4c = 0, and solve for one ratio
23. [A1] Obtain a : b : c = 4 : 1 : − 2, or equivalent
24. [M1] Substitute a relevant point and values of a, b, c in general equation and find d
25. [A1] Obtain correct answer, 4x + y – 2z = 8, or equivalent
26. [M1] OR1: Attempt to calculate vector product of relevant vectors, e.g. (i + 4j + 4k) x (i + 2j + 3k)
27. [A1] Obtain two correct components
28. [A1] Obtain correct answer, e.g. 4i + j – 2k
29. [M1] Substitute a relevant point and find d
30. [A1] Obtain correct answer, 4x + y − 2z = 8, or equivalent
31. [M1] OR2: Using a relevant point and relevant vectors form a 2-parameter equation for the plane
32. [A1] State a correct equation, e.g. r = 4i + 2j + 5k + λ(i + 4j + 4k) + μ(i + 2j + 3k)
33. [A1] State three correct equations in x, y, z, λ and μ
34. [M1] Eliminate λ and μ
35. [A1] Obtain correct answer 4x + y – 2z = 8, or equivalent