A particle moves in a straight line. It starts from rest from a fixed point O on the line. Its velocity at time t s after leaving O is v m s⁻¹, where v = t³ – 8t² + 10t.

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2021). The mark or mark type is shown in brackets.

1. [1] For attempting to integrate v.
2. [1] [s=]1/3 t^3 - 16/5 t^2 + 5t^2 [+C]
3. [1] Use of limit at t = 0 may be implied.
4. [1] Allow displacement = 32/15
5. [1] For attempting to differentiate v.
6. [1] [a=]2t-12t²+10 Allow unsimplified.
7. [1] Dependent on *M1. Set a = 0 and attempt to solve their 3 term equation in √t or t or p(=√t) by treating it as a quadratic equation.
8. [1] Both correct.
9. [1] Dependent on *DM1. Determine the nature of the stationary point by: Either differentiating a and testing the sign of da/dt or by substituting values either side of their t value(s) and attempt to determine the nature of the stationary point(s). If using da/dt then must evaluate it at a t value for M1. Allow use with any t value from their 'quadratic'.
10. [1] Or by using a convincing argument to show that t = 25 gives a minimum value of v. If evaluated then da/dt must be 0.8.
11. [1] AG This mark is awarded only if the previous 6 marks are awarded.