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]