A particle P moves in a straight line such that, t seconds after passing through a fixed point O, its displacement, s metres, is given by s = (2t+1)³ / (t+1) - 1.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2022). The mark or mark type is shown in brackets.

1. [1] (a) M1 for a correct attempt at differentiation using the quotient rule (or product rule if written as (2t+1)³(t+1)⁻¹)
2. [1] (a) A1 for correct derivative of (2t+1)³, i.e. 6(2t+1)²
3. [1] (a) A1 for correctly applying quotient rule: `(t+1) * 6(2t+1)² - (2t+1)³ * 1` in the numerator
4. [1] (a) M1 For factorising out (2t+1)² and simplifying the remaining bracket `[6(t+1) - (2t+1)]`
5. [1] (a) A1 for a=5 and b=4
6. [1] (b) P is never at instantaneous rest because (2t+1)² = 0 implies t = -1/2 (not for t > 0) and (4t+5) = 0 implies t = -5/4 (not for t > 0). Therefore, velocity is never zero for t > 0.