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Date	May 2018	Marks available	5	Reference code	18M.2.SL.TZ1.S_10
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 1
Command term	Find	Question number	S_10	Adapted from	N/A

Question

Let $f\left( x \right) = 12\,\,{\text{cos}}\,x - 5\,\,{\text{sin}}\,x,\,\, - \pi \leqslant x \leqslant 2\pi$ , be a periodic function with $f\left( x \right) = f\left( {x + 2\pi } \right)$

The following diagram shows the graph of $f$ .

There is a maximum point at A. The minimum value of $f$ is −13 .

A ball on a spring is attached to a fixed point O. The ball is then pulled down and released, so that it moves back and forth vertically.

The distance, d centimetres, of the centre of the ball from O at time t seconds, is given by

$d\left( t \right) = f\left( t \right) + 17,\,\,0 \leqslant t \leqslant 5.$

Find the coordinates of A.

[2]

a.

For the graph of $f$ , write down the amplitude.

[1]

b.i.

For the graph of $f$ , write down the period.

[1]

b.ii.

Hence, write $f\left( x \right)$ in the form $p\,\,{\text{cos}}\,\left( {x + r} \right)$ .

[3]

c.

Find the maximum speed of the ball.

[3]

d.

Find the first time when the ball’s speed is changing at a rate of 2 cm s⁻².

[5]

e.

Markscheme

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

−0.394791,13

A(−0.395, 13) A1A1 N2

[2 marks]

a.

13 A1 N1

[1 mark]

b.i.

${2\pi }$ , 6.28 A1 N1

[1 mark]

b.ii.

valid approach (M1)

eg recognizing that amplitude is p or shift is r

$f\left( x \right) = 13\,\,{\text{cos}}\,\left( {x + 0.395} \right)$ (accept p = 13, r = 0.395) A1A1 N3

Note: Accept any value of r of the form $0.395 + 2\pi k,\,\,k \in \mathbb{Z}$

[3 marks]

c.

recognizing need for d ′(t) (M1)

eg −12 sin(t) − 5 cos(t)

correct approach (accept any variable for t) (A1)

eg −13 sin(t + 0.395), sketch of d′, (1.18, −13), t = 4.32

maximum speed = 13 (cms⁻¹) A1 N2

[3 marks]

d.

recognizing that acceleration is needed (M1)

eg a(t), d "(t)

correct equation (accept any variable for t) (A1)

eg $a\left( t \right) = - 2,\,\,\left| {\frac{{\text{d}}}{{{\text{d}}t}}\left( {d'\left( t \right)} \right)} \right| = 2,\,\, - 12\,\,{\text{cos}}\,\left( t \right) + 5\,\,{\text{sin}}\,\left( t \right) = - 2$

valid attempt to solve their equation (M1)

eg sketch, 1.33

1.02154

1.02 A2 N3

[5 marks]

e.

Examiners report

[N/A]

a.

[N/A]

b.i.

[N/A]

b.ii.

[N/A]

c.

[N/A]

d.

[N/A]

e.

Syllabus sections

Topic 2—Functions » AHL 2.9—HL modelling functions

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Topic 2—Functions