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Date May 2019 Marks available 3 Reference code 19M.2.SL.TZ1.S_8
Level Standard Level Paper Paper 2 Time zone Time zone 1
Command term Find Question number S_8 Adapted from N/A

Question

Let  f ( x ) = 2 sin ( 3 x ) + 4 for  x R .

Let  g ( x ) = 5 f ( 2 x ) .

The function g can be written in the form g ( x ) = 10 sin ( b x ) + c .

The range of f is k f ( x ) m . Find k and m .

[3]
a.

Find the range of g .

[2]
b.

Find the value of b and of c .

[3]
c.i.

Find the period of g .

[2]
c.ii.

The equation  g ( x ) = 12  has two solutions where  π  ≤  x  ≤  4 π 3 . Find both solutions.

[3]
d.

Markscheme

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

valid attempt to find range   (M1)

eg  ,  max = 6   min = 2,

2 sin ( 3 × π 6 ) + 4 and  2 sin ( 3 × π 2 ) + 4 ,    2 ( 1 ) + 4 and  2 ( 1 ) + 4 ,

k = 2 m = 6       A1A1 N3

[3 marks]

a.

10 ≤  y ≤ 30      A2 N2

[2 marks]

b.

evidence of substitution (may be seen in part (b))       (M1)

eg    5 ( 2 sin ( 3 ( 2 x ) ) + 4 ) 3 ( 2 x )  

b = 6 c = 20    (accept  10 sin ( 6 x ) + 20 )     A1A1 N3

Note: If no working shown, award N2 for one correct value.

[3 marks]

c.i.

correct working      (A1)

eg   2 π b

1.04719

2 π 6 ( = π 3 ) , 1.05     A1 N2

[2 marks]

c.ii.

valid approach     (M1)

eg    si n 1 ( 8 10 ) 6 x = 0.927 0.154549 x = 0.678147

Note: Award M1 for any correct value for x or 6 x which lies outside the domain of f .

3.81974,  4.03424

x = 3.82 ,   x = 4.03   (do not accept answers in degrees)     A1A1 N3

[3 marks]

d.

Examiners report

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

Syllabus sections

Topic 3— Geometry and trigonometry » SL 3.7—Circular functions: graphs, composites, transformations
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Topic 3— Geometry and trigonometry » SL 3.8—Solving trig equations
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