10 The equations of the lines l and m are given by l: r = (3i - 2j + k) + λ(3i + j + 2k) and m: r = (6i - 3j + 6k) + μ(-2i + 4j + ck), where c is a positive constant. It is given that the angle between l and m is 60°. (a) Find the value of c. (b) Show that the length of the perpendicular from (6, −3, 6) to l is √11.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2023). The mark or mark type is shown in brackets.

1. M1 Use the correct process to calculate the scalar product of the direction vectors (-2+4+2c).
2. M1 Divide the scalar product by the product of the moduli and equate the result to cos 60° Or equivalent e.g. 2+2c=√6√20+c² cos 60° . Allow for the correct process using 60° but the wrong vectors.
3. A1 Obtain correct equation in c e.g. (2+2c)/(√6√(20+c²)) = ½ or 10c² +32c-104=0.
4. A1 Obtain c = 2 Only.
5. B1 Calling (6, -3, 6) A, find AP for a general point P on l e.g. (-3+λ, 1+λ, -5+2λ).
6. *M1 Equate the scalar product of their AP and a direction vector for l to zero and obtain an equation in λ e.g. (-3+λ)+(1+λ)+(-10+4λ)=0.
7. A1 Solve and obtain λ = 2
8. DM1 Carry out a method to calculate |AP| e.g. (-1)² +3² +(-1)² or 1² +3² +1² .
9. A1 AG Obtain √11 from correct working