Write down the first two times ${t_1},{\text{ }}{t_2} > 0$, when the particle changes direction.

(i)     Find the time $t < {t_2}$ when the particle has a maximum velocity.

(ii)     Find the time $t < {t_2}$ when the particle has a minimum velocity.

Find the distance travelled by the particle between times $t = {t_1}$ and $t = {t_2}$.

${t_1} = 1.77{\text{ (s)}}\;\;\;\left( { = \sqrt \pi  {\text{ (s)}}} \right)\;\;\;{\text{and}}\;\;\;{t_2} = 2.51{\text{ (s)}}\;\;\;\left( { = \sqrt {2\pi } {\text{ (s)}}} \right)$     A1A1

[2 marks]

(i)     attempting to find (graphically or analytically) the first ${t_{\max }}$     (M1)

$t = 1.25{\text{ (s)}}\;\;\;\left( { = \sqrt {\frac{\pi }{2}} {\text{ (s)}}} \right)$     A1

(ii)     attempting to find (graphically or analytically) the first ${t_{\min }}$     (M1)

$t = 2.17{\text{ (s)}}\;\;\;\left( { = \sqrt {\frac{{3\pi }}{2}} {\text{ (s)}}} \right)$     A1

[4 marks]

distance travelled $= \left| {\int_{1.772 \ldots }^{2.506 \ldots } {1 - {{\text{e}}^{ - \sin {t^2}}}{\text{d}}t} } \right|\;\;\;$(or equivalent)     (M1)

$= 0.711{\text{ (m)}}$     A1

Note:     Award M1 for attempting to form a definite integral involving $1 - {{\text{e}}^{ - \sin {t^2}}}$. To award the A1, correct limits leading to $0.711$ must include the use of absolute value or a statement such as “distance must be positive”.

In part (c), award A1FT for a candidate working in degree mode $\left( {5.39{\text{ (m)}}} \right)$.

[2 marks]

Total [8 marks]