The line l has equation r = 5i – 3j – k + λ(i – 2j + k). The plane p has equation (r - i - 2j).(3i + j + k) = 0. The line l intersects the plane p at the point A.

Mark scheme

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

1. [M1*] Substitute for r and expand the scalar product to obtain an equation in λ
2. [M1(dep*)] Solve a linear equation for λ
3. [A1] Obtain λ = -3 and position vector rA= 2i +3j – 4k for A
4. [B1] State or imply a normal vector of p is 3i + j + k, or equivalent
5. [M1] Use correct method to evaluate a scalar product of relevant vectors e.g. (i- 2j + k).(3i + j + k)
6. [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
7. [A1] Obtain answer 14.3° or 0.249 radians
8. [B1] Taking the direction vector of the line to be ai + bj + ck, state a relevant equation in a, b, c, e.g. 3a + b + c = 0
9. [M1] State a second relevant equation, e.g. a – 2b + c = 0, and solve for one ratio, e.g. a : b
10. [A1] Obtain a : b : c = 3 : - 2 : - 7, or equivalent
11. [A1ft] State answer r = 2i + 3j – 4k + µ (3i – 2j – 7k)