The line l has equation r = i + 3j – 2k + λ(i – 2j + 3k). The plane p has equation 2x + y − 3z = 5.

Mark scheme

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

1. [B1] Express general point of l in component form e.g. (1 + λ, 3 – 2λ, – 2 + 3λ)
2. [M1] Substitute in equation of p and solve for λ
3. [A1] Obtain final answer (5/3)i + (5/3)j from λ = 2/3
4. [M1] Use correct method to evaluate a scalar product of relevant vectors e.g. (i - 2j + 3k).(2i + j - 3k)
5. [M1] Using the correct process for calculating the moduli, divide the scalar product by the product of the moduli and evaluate the inverse sine or cosine of the result
6. [A1] Obtain answer 40.0° or 0.698 radians
7. [M1] Use correct method to evaluate a vector product of relevant vectors e.g. (i - 2j + 3k)x(2i + j - 3k)
8. [M1] Using the correct process for calculating the moduli, divide the modulus of the vector product by the product of the moduli of the two vectors and evaluate the inverse sine or cosine of the result
9. [A1] Obtain answer 40.0° or 0.698 radians
10. [B1] State a ⋅ n_q = 0 or p ⋅ n_q = 0 implies 2a + b - 3c = 0 or 2a + b - 3c = 0
11. [M1] Obtain two relevant equations and solve for one ratio, e.g. a:b
12. [A1] Obtain a:b:c = 3:9:5
13. [M1] Substitute a, b, c and a relevant point in the plane equation and evaluate d
14. [A1] Obtain answer 3x + 9y + 5z = 20
15. [M1] Attempt to calculate vector product of relevant vectors, e.g. (i - 2j + 3k) × (2i + j - 3k)
16. [A1] Obtain two correct components
17. [A1] Obtain correct answer, e.g. 3i + 9j + 5k
18. [M1] Use the product and a relevant point to find d
19. [A1] Obtain answer 3x + 9y + 5z = 20, or equivalent
20. [M1] Attempt to form a 2-parameter equation with relevant vectors
21. [A1] State a correct equation e.g. r = i + 3j – 2k + λ(i – 2j + 3k) + μ(2i + j – 3k)
22. [A1] State 3 equations in x, y, z, λ and μ
23. [M1] Eliminate λ and μ
24. [A1] Obtain answer 3x + 9y + 5z = 20