Write down the values of $t$ when $a = 0$.

Hence or otherwise, find all possible values of $t$ for which the velocity of P is decreasing.

Find an expression for the velocity of P at time $t$.

Find the total distance travelled by P when its velocity is increasing.

$t = \frac{2}{3}{\text{ (exact), }}0.667,{\text{ }}t = 4$     A1A1     N2

[2 marks]

recognizing that $v$ is decreasing when $a$ is negative     (M1)

eg$\,\,\,\,\,$$a < 0,{\text{ }}3{t^2} - 14t + 8 \leqslant 0$, sketch of $a$

correct interval     A1     N2

eg$\,\,\,\,\,$$\frac{2}{3} < t < 4$

[2 marks]

valid approach (do not accept a definite integral)     (M1)

eg$\,\,\,\,\,$$v\int a$

correct integration (accept missing $c$)     (A1)(A1)(A1)

${t^3} - 7{t^2} + 8t + c$

substituting $t = 0,{\text{ }}v = 3$ , (must have $c$)     (M1)

eg$\,\,\,\,\,$$3 = {0^3} - 7({0^2}) + 8(0) + c,{\text{ }}c = 3$

$v = {t^3} - 7{t^2} + 8t + 3$     A1     N6

[6 marks]

recognizing that $v$ increases outside the interval found in part (b)     (M1)

eg$\,\,\,\,\,$$0 < t < \frac{2}{3},{\text{ }}4 < t < 5$, diagram

one correct substitution into distance formula     (A1)

eg$\,\,\,\,\,$$\int_0^{\frac{2}{3}} {\left| v \right|,{\text{ }}\int_4^5 {\left| v \right|} ,{\text{ }}\int_{\frac{2}{3}}^4 {\left| v \right|} ,{\text{ }}\int_0^5 {\left| v \right|} }$

one correct pair     (A1)

eg$\,\,\,\,\,$3.13580 and 11.0833, 20.9906 and 35.2097

14.2191     A1     N2

$d = 14.2{\text{ (m)}}$

[4 marks]