| Date | May 2018 | Marks available | 3 | Reference code | 18M.2.sl.TZ2.9 |
| Level | SL only | Paper | 2 | Time zone | TZ2 |
| Command term | Write down | Question number | 9 | Adapted from | N/A |
Question
A particle P moves along a straight line. The velocity v m s−1 of P after t seconds is given by v (t) = 7 cos t − 5t cos t, for 0 ≤ t ≤ 7.
The following diagram shows the graph of v.
Find the initial velocity of P.
Find the maximum speed of P.
Write down the number of times that the acceleration of P is 0 m s−2 .
Find the acceleration of P when it changes direction.
Find the total distance travelled by P.
Markscheme
initial velocity when t = 0 (M1)
eg v(0)
v = 17 (m s−1) A1 N2
[2 marks]
recognizing maximum speed when \(\left| v \right|\) is greatest (M1)
eg minimum, maximum, v' = 0
one correct coordinate for minimum (A1)
eg 6.37896, −24.6571
24.7 (ms−1) A1 N2
[3 marks]
recognizing a = v ′ (M1)
eg \(a = \frac{{{\text{d}}v}}{{{\text{d}}t}}\), correct derivative of first term
identifying when a = 0 (M1)
eg turning points of v, t-intercepts of v ′
3 A1 N3
[3 marks]
recognizing P changes direction when v = 0 (M1)
t = 0.863851 (A1)
−9.24689
a = −9.25 (ms−2) A2 N3
[4 marks]
correct substitution of limits or function into formula (A1)
eg \(\int_0^7 {\left| {\,v\,} \right|,\,\int_0^{0.8638} {v{\text{d}}t - \int_{0.8638}^7 {v{\text{d}}t} } ,\,\,\int {\left| {\,7\,{\text{cos}}\,x - 5{x^{{\text{cos}}\,x}}\,} \right|} \,dx,\,\,3.32 = 60.6} \)
63.8874
63.9 (metres) A2 N3
[3 marks]
