A particle moves in a straight line. The velocity v ms⁻¹ of the particle t s after leaving a fixed point O is given by v = k(20+pt-6t²), where k and p are constants. The acceleration of the particle at t = 1 is 42 ms⁻², and the displacement of the particle from O at t = 1 is 93 m.

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2025). The mark or mark type is shown in brackets.

1. *M1 Decrease power by 1 and a change in coefficient in at least one term (which must be the same term); a = v/t is M0
2. A1 OE; Allow unsimplified.
3. *M1 Increase power by 1 and a change in coefficient in at least one term (which must be the same term) s = vt is M0
4. A1 OE; Allow unsimplified.
5. DM1 Dependent on both previous M marks. Allow sign errors only in solving. Must be solving the correct equations. Must have c=0 if evaluated. Must get to 'p =' or ‘k='or attempt to verify for both equations. Working must be seen for this mark. Must see at least one line of working once either p or k have been eliminated.
6. A1 AG Any error seen is A0
7. *M1 Using their 2 term linear a that has come from differentiation to solve for t, which must be positive, using correct p. OE e.g. 26/12.
8. *M1 Attempt to solve given quadratic expression equated to 0 using correct p and k. Must get at least 1 t value.
9. A1 If 2 values given, they must be both correct.
10. DM1 Dependent on previous 2 M marks. For using their positive limits correctly in their s which has come from integration. May be implied by correct answer.
11. A1 Allow 4913/18 272.944... SCB1 for the last 2 marks for answer without seeing 3(20t+13t² - 2t³).