The equations of two straight lines are r=i+j+2ak+λ(3i+4j+ak) and r=-3i-j+4k+μ(−i+2j+2k) where a is a constant.

Mark scheme

Marking points from the official mark scheme (May/June 2024). The mark or mark type is shown in brackets.

1. [M1] Carry out correct process for evaluating the scalar product of direction vectors
2. [M1] Using the correct process for the moduli, divide the scalar product by the product of the moduli and equate to ± √2/2
3. [A1] State a correct equation in any form, e.g. (5+2a) / (√(25+a²)√9) = [±]√2/2
4. [DM1] Form a quadratic equation in a with 3 or more terms all on one side and solve for a.
5. [A2] Obtain a = 5 and a = 35
6. [B1] Express general point of at least one line correctly in component form,
7. [M1] Equate at least two pairs of corresponding components and solve for λ or μ or a
8. [A1] Obtain λ = −1 or μ = −1
9. [A1] Obtain a = 2
10. [A1] Obtain position vector of the point of intersection is –2i – 3j + 2k