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Date May 2018 Marks available 4 Reference code 18M.2.sl.TZ2.9
Level SL only Paper 2 Time zone TZ2
Command term Find 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.

[2]
a.

Find the maximum speed of P.

[3]
b.

Write down the number of times that the acceleration of P is 0 m s−2 .

[3]
c.

Find the acceleration of P when it changes direction.

[4]
d.

Find the total distance travelled by P.

[3]
e.

Markscheme

initial velocity when t = 0      (M1)

eg v(0)

v = 17 (m s−1)      A1 N2

[2 marks]

a.

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]

b.

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]

c.

recognizing P changes direction when = 0       (M1)

t = 0.863851      (A1)

−9.24689

a = −9.25 (ms−2)      A2 N3

[4 marks]

d.

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]

e.

Examiners report

[N/A]
a.
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b.
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c.
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d.
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e.

Syllabus sections

Topic 6 - Calculus » 6.6 » Kinematic problems involving displacement \(s\), velocity \(v\) and acceleration \(a\).
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