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Date November 2021 Marks available 5 Reference code 21N.1.SL.TZ0.7
Level Standard Level Paper Paper 1 (without calculator) Time zone Time zone 0
Command term Show that Question number 7 Adapted from N/A

Question

A particle P moves along the x-axis. The velocity of P is vms-1 at time t seconds, where v(t)=4+4t-3t2 for 0t3. When t=0, P is at the origin O.

Find the value of t when P reaches its maximum velocity.

[2]
a.i.

Show that the distance of P from O at this time is 8827 metres.

[5]
a.ii.

Sketch a graph of v against t, clearly showing any points of intersection with the axes.

[4]
b.

Find the total distance travelled by P.

[5]
c.

Markscheme

valid approach to find turning point (v'=0, -b2a, average of roots)                 (M1)

4-6t=0   OR   -42-3   OR   -23+22

t=23 (s)                 A1

  

[2 marks]

a.i.

attempt to integrate v                 (M1)

vdt=4+4t-3t2 dt=4t+2t2-t3+c                 A1A1


Note: Award A1 for 4t+2t2, A1 for -t3.


attempt to substitute their t into their solution for the integral                 (M1)

distance=423+2232-233

=83+89-827 (or equivalent)                           A1

=8827 (m)                   AG

  

[5 marks]

a.ii.

valid approach to solve 4+4t-3t2=0   (may be seen in part (a))                 (M1)

2-t2+3t  OR  -4±16+48-6

correct x- intercept on the graph at t=2                 A1


Note: The following two A marks may only be awarded if the shape is a concave down parabola. These two marks are independent of each other and the (M1).


correct domain from 0 to 3 starting at (0,4)                 A1


Note: The 3 must be clearly indicated.


vertex in approximately correct place for t=23 and v>4                 A1

 

[4 marks]

b.

recognising to integrate between 0 and 2, or 2 and 3   OR   034+4t-3t2dt                (M1)

024+4t-3t2dt

=8                 A1

234+4t-3t2dt

=-5                 A1

valid approach to sum the two areas (seen anywhere)                (M1)

02vdt-23vdt   OR   02vdt+23vdt

total distance travelled =13 (m)                 A1

 

[5 marks]

c.

Examiners report

[N/A]
a.i.
[N/A]
a.ii.
[N/A]
b.
[N/A]
c.

Syllabus sections

Topic 5 —Calculus » SL 5.5—Integration introduction, areas between curve and x axis
Show 162 related questions
Topic 2—Functions » SL 2.6—Quadratic function
Topic 5 —Calculus » SL 5.9—Kinematics problems
Topic 2—Functions
Topic 5 —Calculus

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