The line l has equation r = i + 2j + 3k + μ(2i – j – 2k).

Mark scheme

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

1. [B1] Find PQ for a general point Q on l, e.g. –3i+6k+μ(2i−j−2k)
2. [M1] Calculate scalar product of PQ and a direction vector for l and equate the result to zero
3. [A1] Solve for μ and obtain μ = 2
4. [M1] Carry out a complete method for finding the length of PQ
5. [A1] Obtain answer 3
6. [B1] Calling the point (1, 2, 3) A, state AP (or PA) in component form, e.g. 3i – 6k
7. [M1] Use a scalar product with a direction vector for l to find the projection of AP (or PA) on l
8. [A1] Obtain correct answer in any form, e.g. 18/√9
9. [M1] Use Pythagoras to find the perpendicular
10. [A1] Obtain answer 3
11. [B1] State AP (or PA) in component form
12. [M1] Calculate a vector product with a direction vector for l
13. [A1] Obtain correct answer, e.g. 6i – 6j – 3k
14. [M1] Divide modulus of the product by that of the direction vector
15. [A1] Obtain answer 3
16. [M1] Substitute coordinates of a general point of l in the plane equation and equate constant terms
17. [A1] Obtain a correct equation, e.g. a + 2b + 6 = 13
18. [M1] Equate the coefficient of μ to zero
19. [A1] Obtain a correct equation, e.g. 2a - b − 4 = 0
20. [M1] Substitute (1, 2, 3) in the plane equation
21. [A1] Obtain a correct equation, e.g. a + 2b + 6 = 13
22. [M1] Find a second point on l and obtain an equation in a and/or b
23. [A1] Obtain a correct equation, e.g. 5a – 2 = 13
24. [M1] Equate scalar product of a direction vector for l and a vector normal for the plane to zero
25. [A1] Obtain a correct equation, e.g. 2a – b – 4 = 0
26. [M1] Solve for a or for b
27. [A1] Obtain a = 3 and b = 2