| Date | November 2015 | Marks available | 4 | Reference code | 15N.2.hl.TZ0.9 |
| Level | HL only | Paper | 2 | Time zone | TZ0 |
| Command term | Find | Question number | 9 | Adapted from | N/A |
Question
A particle can move along a straight line from a point \(O\). The velocity \(v\), in \({\text{m}}{{\text{s}}^{ - 1}}\), is given by the function \(v(t) = 1 - {{\text{e}}^{ - \sin {t^2}}}\) where time \(t \ge 0\) is measured in seconds.
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 \(t = {t_2}\).
Markscheme
\({t_1} = 1.77{\text{ (s)}}\;\;\;\left( { = \sqrt \pi {\text{ (s)}}} \right)\;\;\;{\text{and}}\;\;\;{t_2} = 2.51{\text{ (s)}}\;\;\;\left( { = \sqrt {2\pi } {\text{ (s)}}} \right)\) 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]
