A particle travels in a straight line. The velocity of the particle at time t s after leaving a point O is v ms⁻¹, where v = kt²-4t+3. The distance travelled by the particle in the first 2 s of its motion is 6m. You may assume that v ≥ 0 in the first 2 s of its motion.

Mark scheme

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

1. [M1*] For attempt at integration. The power of t must increase by 1 with a change of coefficient in the same term. Use of s = vt scores M0.
2. [A1] 1/3 kt³ - 2t² + 3t [Allow unsimplified.]
3. [DM1] Use of limits 0 and 2 with 6 to form an equation in k only (without c but allow with +c-c).
4. [A1] k=3
5. [M1] For attempt at differentiation. Must have expression of the form at+b with a ≠ 3, unless their k = 1/2. Allow 2kt –4.
6. [A1FT] t = 2/3 [FT their k t = 2/their k. Allow without working.]
7. [A1] v = 5/3 ms⁻¹ [Allow 1.67 or better for v.]