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Date	May 2017	Marks available	6	Reference code	17M.2.SL.TZ2.S_7
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 2
Command term	Find and Justify	Question number	S_7	Adapted from	N/A

Question

Note: In this question, distance is in metres and time is in seconds.

A particle moves along a horizontal line starting at a fixed point A. The velocity $v$ of the particle, at time $t$ , is given by $v(t) = \frac{{2{t^2} - 4t}}{{{t^2} - 2t + 2}}$ , for $0 \leqslant t \leqslant 5$ . The following diagram shows the graph of $v$

M17/5/MATME/SP2/ENG/TZ2/07

There are $t$ -intercepts at $(0,{\text{ }}0)$ and $(2,{\text{ }}0)$ .

Find the maximum distance of the particle from A during the time $0 \leqslant t \leqslant 5$ and justify your answer.

Markscheme

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

METHOD 1 (displacement)

recognizing $s = \int {v{\text{d}}t}$ (M1)

consideration of displacement at $t = 2$ and $t = 5$ (seen anywhere) M1

eg $\,\,\,\,\,$ $\int_0^2 v$ and $\int_0^5 v$

Note: Must have both for any further marks.

correct displacement at $t = 2$ and $t = 5$ (seen anywhere) A1A1

$- 2.28318$ (accept 2.28318), 1.55513

valid reasoning comparing correct displacements R1

eg $\,\,\,\,\,$ $\left| { - 2.28} \right| > \left| {1.56} \right|$ , more left than right

2.28 (m) A1 N1

Note: Do not award the final A1 without the R1.

METHOD 2 (distance travelled)

recognizing distance $= \int {\left| v \right|{\text{d}}t}$ (M1)

consideration of distance travelled from $t = 0$ to 2 and $t = 2$ to 5 (seen anywhere) M1

eg $\,\,\,\,\,$ $\int_0^2 v$ and $\int_2^5 v$

Note: Must have both for any further marks

correct distances travelled (seen anywhere) A1A1

2.28318, (accept $- 2.28318$ ), 3.83832

valid reasoning comparing correct distance values R1

eg $\,\,\,\,\,$ $3.84 - 2.28 < 2.28,{\text{ }}3.84 < 2 \times 2.28$

2.28 (m) A1 N1

Note: Do not award the final A1 without the R1.

[6 marks]

Examiners report

[N/A]

Syllabus sections

Topic 5—Calculus » AHL 5.13—Kinematic problems

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