The planes m and n have equations 3x + y − 2z = 10 and x − 2y + 2z = 5 respectively. The line l has equation r = 4i + 2j + k + λ(i + j + 2k).

Mark scheme

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

1. [M1] (i) EITHER: Expand scalar product of a normal to m and a direction vector of l
2. [A1] (i) Verify scalar product is zero
3. [A1] (i) Verify that one point of l does not lie in the plane
4. [M1] (i) OR: Substitute coordinates of a general point of l in the equation of the plane m
5. [A1] (i) Obtain correct equation in λ in any form
6. [A1] (i) Verify that the equation is not satisfied for any value of λ
7. [M1] (ii) Use correct method to evaluate a scalar product of normal vectors to m and n
8. [M1] (ii) Using the correct process for the moduli, divide the scalar product by the product of the moduli and evaluate the inverse cosine of the result
9. [A1] (ii) Obtain answer 74.5° or 1.30 radians
10. [M1] (iii) EITHER: Using the components of a general point P of l form an equation in λ by equating the perpendicular distance from n to 2
11. [M1] (iii) OR: Take a point Q on l, e.g. (5, 3, 3) and form an equation in λ by equating the length of the projection of QP onto a normal to plane n to 2
12. [A1] (iii) Obtain a correct modular or non-modular equation in any form
13. [A1] (iii) Solve for λ and obtain a position vector for P, e.g. 7i + 5j + 7k from λ = 3
14. [A1] (iii) Obtain position vector of the second point, e.g. 3i + j – k from λ = –1