User interface language: English | Español

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 ( x ) = 12 cos x 5 sin x , π x 2 π , be a periodic function with  f ( x ) = f ( x + 2 π )

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 ( t ) = f ( t ) + 17 , 0 t 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 ( x ) in the form  p cos ( x + r ) .

[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 π , 6.28      A1 N1

[1 mark]

b.ii.

valid approach      (M1)

eg recognizing that amplitude is p or shift is r

f ( x ) = 13 cos ( x + 0.395 )    (accept p = 13, r = 0.395)     A1A1 N3

Note: Accept any value of r of the form  0.395 + 2 π k , k 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 ( t ) = 2 , | d d t ( d ( t ) ) | = 2 , 12 cos ( t ) + 5 sin ( t ) = 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 5 —Calculus » SL 5.6—Differentiating polynomials n E Q. Chain, product and quotient rules
Show 127 related questions
Topic 3— Geometry and trigonometry » SL 3.7—Circular functions: graphs, composites, transformations
Topic 5 —Calculus » SL 5.8—Testing for max and min, optimisation. Points of inflexion
Topic 5 —Calculus » SL 5.9—Kinematics problems
Topic 3— Geometry and trigonometry
Topic 5 —Calculus

View options