In this question i is a unit vector in the positive x-direction and j is a unit vector in the positive y-direction. Time is in seconds and distances are in metres. The diagram shows the initial positions and velocities of two particles, A and B, that move in the x-y plane. Particle A starts from the origin O at time t = 0. It moves with constant speed 10 ms⁻¹ in the direction 60° above the x-axis.

Mark scheme

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

1. [B2] x = 5 y = 5√3 [B1] for either component correct
2. [B1] [0,0] + t [5, 5√3] oe, isw [FT] their [5, 5√3] which must be a vector with at least one non-zero component
3. [B2] [2√3, 9] + t [5/3, 0] oe, isw [B1] for x component 2√3 + (5/3)t seen or y component 9 seen or [2√3, 9] + t[5/3, 0] with at most one error but must include t
4. [M1] 5t√3 = 9 or 5t = 2√3 + (5/3)t [FT] their position vector of A and their position vector of B providing • both are in terms of t and • at least one is of form [a;b]+t[c;d] where a, b, c and d are constants
5. [A2] Shows the exact times to be the same e.g. t = 9/(5√3) = 3√3/5 and t = 2√3 / (10/3) = 3√3/5 oe [A1] for t = 9/(5√3) and t = 2√3 / (10/3) oe
6. OR [A2] Finds a correct value for t, as above, and shows this satisfies the other equation
7. OR [A2] Finds a correct value for t, as above, and shows both particles are at [√3; 9] oe at this time