Q337 · Practice questionTechnology-freeComplex unfamiliar8 marks
QUESTION 337 (8 marks)
A robot's position is metres for seconds. An inspection occurs at a random time uniformly distributed on .
![Position graph s(t)=t cubed-6t squared+9t on [0,4], with a rise, a fall back to zero and another rise.](/qce-data/assets/methods-2026-MM26-ORIG-099-mobile.svg)
a)[2 marks]
Determine the times when the robot changes direction.
b)[3 marks]
Determine its total distance travelled over the four seconds.
c)[3 marks]
Determine the exact probability that its speed at inspection is no more than m/s.
WORKED SOLUTION
8 marksPractice marking scheme
ANSWER
a) s and s. b) m. c) .
Worked solution
a) changes sign at and .
b) Positions at are . Distance is .
c) . Thus reduces to , or . Uniform probability is .
Exact answers unless rounding is specified; equivalent forms accepted.
a) Differentiates and finds both zeros.
a) Confirms direction changes.
b) Evaluates the positions at boundaries and reversals.
b) Adds the absolute changes.
b) Gives 12 metres.
c) Forms the absolute speed inequality.
c) Determines the time interval.
c) Calculates the exact uniform probability.
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