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Date November 2016 Marks available 2 Reference code 16N.2.SL.TZ0.S_9
Level Standard Level Paper Paper 2 Time zone Time zone 0
Command term Find Question number S_9 Adapted from N/A

Question

A particle P starts from a point A and moves along a horizontal straight line. Its velocity v  cm s 1 after t seconds is given by

v ( t ) = { 2 t + 2 , for  0 t 1 3 t + 4 t 2 7 , for  1 t 12

The following diagram shows the graph of v .

N16/5/MATME/SP2/ENG/TZ0/09

P is at rest when t = 1 and t = p .

When t = q , the acceleration of P is zero.

Find the initial velocity of P .

[2]
a.

Find the value of p .

[2]
b.

(i)     Find the value of q .

(ii)     Hence, find the speed of P when t = q .

[4]
c.

(i)     Find the total distance travelled by P between t = 1 and t = p .

(ii)     Hence or otherwise, find the displacement of P from A when t = p .

[6]
d.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

valid attempt to substitute t = 0 into the correct function     (M1)

eg 2 ( 0 ) + 2

2     A1     N2

[2 marks]

a.

recognizing v = 0  when P is at rest     (M1)

5.21834

p = 5.22   ( seconds )      A1     N2

[2 marks]

b.

(i)     recognizing that a = v      (M1)

eg v = 0 , minimum on graph

1.95343

q = 1.95      A1     N2

(ii)     valid approach to find their minimum     (M1)

eg v ( q ) ,   1.75879 , reference to min on graph

1.75879

speed = 1.76   ( c m s 1 )      A1     N2

[4 marks]

c.

(i)     substitution of correct  v ( t ) into distance formula,     (A1)

eg 1 p | 3 t + 4 t 2 7 | d t ,   | 3 t + 4 t 2 7 d t |

4.45368

distance = 4.45   ( cm )      A1     N2

(ii)     displacement from t = 1 to t = p (seen anywhere)     (A1)

eg 4.45368 ,   1 p ( 3 t + 4 t 2 7 ) d t

displacement from t = 0 to t = 1     (A1)

eg 0 1 ( 2 t + 2 ) d t ,   0.5 × 1 × 2 ,  1

valid approach to find displacement for 0 t p     M1

eg 0 1 ( 2 t + 2 ) d t + 1 p ( 3 t + 4 t 2 7 ) d t ,   0 1 ( 2 t + 2 ) d t 4.45

3.45368

displacement = 3.45   ( cm )      A1     N2

[6 marks]

d.

Examiners report

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

Syllabus sections

Topic 5—Calculus » AHL 5.13—Kinematic problems
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