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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 \(f(x) = 6x\sqrt {1 - {x^2}} \), for \( - 1 \leqslant x \leqslant 1\), and \(g(x) = \cos (x)\), for \(0 \leqslant x \leqslant \pi \).

Let \(h(x) = (f \circ g)(x)\).

Write \(h(x)\) in the form \(a\sin (bx)\), where \(a,{\text{ }}b \in \mathbb{Z}\).

[5]
a.

Hence find the range of \(h\).

[2]
b.

Markscheme

attempt to form composite in any order     (M1)

eg\(\,\,\,\,\,\)\(f\left( {g(x)} \right),{\text{ }}\cos \left( {6x\sqrt {1 - {x^2}} } \right)\)

correct working     (A1)

eg\(\,\,\,\,\,\)\(6\cos x\sqrt {1 - {{\cos }^2}x} \)

correct application of Pythagorean identity (do not accept \({\sin ^2}x + {\cos ^2}x = 1\))     (A1)

eg\(\,\,\,\,\,\)\({\sin ^2}x = 1 - {\cos ^2}x,{\text{ }}6\cos x\sin x,{\text{ }}6\cos x \left| \sin x\right|\)

valid approach (do not accept \(2\sin x\cos x = \sin 2x\))     (M1)

eg\(\,\,\,\,\,\)\(3(2\cos x\sin x)\)

\(h(x) = 3\sin 2x\)    A1     N3

[5 marks]

a.

valid approach     (M1)

eg\(\,\,\,\,\,\)amplitude \( = 3\), sketch with max and min \(y\)-values labelled, \( - 3 < y < 3\)

correct range     A1     N2

eg\(\,\,\,\,\,\)\( - 3 \leqslant y \leqslant 3\), \([ - 3,{\text{ }}3]\) 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 \(\cos x\), 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 \(\sin x\) , \(\cos x\) and \(\tan x\) : their domains and ranges; amplitude, their periodic nature; and their graphs.
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