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Date May 2016 Marks available 5 Reference code 16M.1.sl.TZ2.6
Level SL only Paper 1 Time zone TZ2
Command term Write Question number 6 Adapted from N/A

Question

Let f(x)=6x1x2f(x)=6x1x2, for 1x1, and g(x)=cos(x), for 0xπ.

Let h(x)=(fg)(x).

Write h(x) in the form asin(bx), where a, bZ.

[5]
a.

Hence find the range of h.

[2]
b.

Markscheme

attempt to form composite in any order     (M1)

egf(g(x)), cos(6x1x2)

correct working     (A1)

eg6cosx1cos2x

correct application of Pythagorean identity (do not accept sin2x+cos2x=1    (A1)

egsin2x=1cos2x, 6cosxsinx, 6cosx|sinx|

valid approach (do not accept 2sinxcosx=sin2x)     (M1)

eg3(2cosxsinx)

h(x)=3sin2x    A1     N3

[5 marks]

a.

valid approach     (M1)

egamplitude =3, sketch with max and min y-values labelled, 3<y<3

correct range     A1     N2

eg3y3, [3, 3] from 3 to 3

Note:     Do not award A1 for 3<y<3 or for “between 3 and 3”.

[2 marks]

b.

Examiners report

In part (a), nearly all candidates found the correct composite function in terms of cosx, though many did not get any further than this first step in their solution to the question. While some candidates seemed to recognize the need to use trigonometric identities, most were unsuccessful in finding the correct expression in the required form.

a.

In part (b), very few candidates were able to provide the correct range of the function.

b.

Syllabus sections

Topic 3 - Circular functions and trigonometry » 3.4 » The circular functions sinx , cosx and tanx : their domains and ranges; amplitude, their periodic nature; and their graphs.
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