The line l has equation r = 4i + 3j – k + μ(i + 2j – 2k). The plane p has equation 2x – 3y – z = 4.

Mark scheme

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

1. [B1] Express general point of l in component form, e.g. r=(4+ μ)i+(3+2μ)j+(−1−2μ)k, or equivalent NB: Calling the vector a + μb, the B1 is earned by a correct reduction of the sum to a single vector or by expressing the substitution as a distributed sum a.n + μb.n
2. [M1] Substitute in given equation of p and solve for μ
3. [A1] Obtain final answer 2i – j + 3k from μ=−2
4. [M1] Using the correct process, evaluate the scalar product of a direction vector for l and a normal for p
5. [M1] Using the correct process for the moduli, divide the scalar product by the product of the moduli and find the inverse sine or cosine of the result
6. [A1] Obtain answer 10.3° (or 0.179 radians)
7. [B1] EITHER: State a + 2b – 2c = 0 or 2a – 3b – c = 0
8. [M1] Obtain two relevant equations and solve for one ratio, e.g. a : b
9. [A1] Obtain a : b : c = 8 : 3 : 7, or equivalent
10. [M1] Substitute a, b, c and given point and evaluate d
11. [A1] Obtain answer 8x + 3y + 7z = 5) OR1: [M1] Attempt to calculate vector product of relevant vectors, e.g. (2i – 3j – k)×(i + 2j – 2k)
12. [A1] Obtain two correct components of the product
13. [A1] Obtain correct product, e.g. 8i + 3j +7k
14. [M1] Use the product and the given point to find d
15. [A1] Obtain answer 8x + 3y + 7z = 5, or equivalent) OR2: [M1] Attempt to form a 2-parameter equation with relevant vectors
16. [A1] State a correct equation, e.g. r = 4j – k + λ(i + 2j – 2k) + μ (2i – 3j – k)
17. [A1] State 3 equations in x, y, z, λ and μ
18. [M1] Eliminate λ and μ
19. [A1] State answer 8x + 3y + 7z = 5, or equivalent)