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Date	November 2019	Marks available	8	Reference code	19N.1.SL.TZ0.S_6
Level	Standard Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 0
Command term	Find	Question number	S_6	Adapted from	N/A

Question

Let $f\left( x \right) = 4\,{\text{cos}}\left( {\frac{x}{2}} \right) + 1$ , for $0 \leqslant x \leqslant 6\pi$ . Find the values of $x$ for which $f\left( x \right) > 2\sqrt 2 + 1$ .

Markscheme

METHOD 1 – FINDING INTERVALS FOR x

$4\,{\text{cos}}\left( {\frac{x}{2}} \right) + 1 > 2\sqrt 2 + 1$

correct working (A1)

eg $4\,{\text{cos}}\left( {\frac{x}{2}} \right) = 2\sqrt 2$ , ${\text{cos}}\left( {\frac{x}{2}} \right) > \frac{{\sqrt 2 }}{2}$

recognizing ${\text{co}}{{\text{s}}^{ - 1}}\frac{{\sqrt 2 }}{2} = \frac{\pi }{4}$ (A1)

one additional correct value for $\frac{x}{2}$ (ignoring domain and equation/inequalities) (A1)

eg $- \frac{\pi }{4}{\text{, }}\frac{{7\pi }}{4}{\text{, }}315^\circ {\text{, }}\frac{{9\pi }}{4}{\text{, }} - 45^\circ {\text{, }}\frac{{15\pi }}{4}$

three correct values for $x$ A1A1

eg $\frac{\pi }{2}{\text{, }}\frac{{7\pi }}{2}{\text{, }}\frac{{9\pi }}{2}$

valid approach to find intervals (M1)

eg

correct intervals (must be in radians) A1A1 N2

$0 \leqslant x < \frac{\pi }{2}$ , $\frac{{7\pi }}{2} < x < \frac{{9\pi }}{2}$

Note: If working shown, award A1A0 if inclusion/exclusion of endpoints is incorrect. If no working shown award N1.
If working shown, award A1A0 if both correct intervals are given, and additional intervals are given. If no working shown award N1.
Award A0A0 if inclusion/exclusion of endpoints are incorrect and additional intervals are given.

METHOD 2 – FINDING INTERVALS FOR $\frac{x}{2}$

$4\,{\text{cos}}\left( {\frac{x}{2}} \right) + 1 > 2\sqrt 2 + 1$

correct working (A1)

eg $4\,{\text{cos}}\left( {\frac{x}{2}} \right) = 2\sqrt 2$ , ${\text{cos}}\left( {\frac{x}{2}} \right) > \frac{{\sqrt 2 }}{2}$

recognizing ${\text{co}}{{\text{s}}^{ - 1}}\frac{{\sqrt 2 }}{2} = \frac{\pi }{4}$ (A1)

one additional correct value for $\frac{x}{2}$ (ignoring domain and equation/inequalities) (A1)

eg $- \frac{\pi }{4}{\text{, }}\frac{{7\pi }}{4}{\text{, }}315^\circ {\text{, }}\frac{{9\pi }}{4}{\text{, }} - 45^\circ {\text{, }}\frac{{15\pi }}{4}$

three correct values for $\frac{x}{2}$ A1

eg $\frac{\pi }{4}{\text{, }}\frac{{7\pi }}{4}{\text{, }}\frac{{9\pi }}{4}$

valid approach to find intervals (M1)

eg

one correct interval for $\frac{x}{2}$ A1

eg $0 \leqslant \frac{x}{2} < \frac{\pi }{4}{\text{, }}\frac{{7\pi }}{4} < \frac{x}{2} < \frac{{9\pi }}{4}$

correct intervals (must be in radians) A1A1 N2

$0 \leqslant x < \frac{\pi }{2}$ , $\frac{{7\pi }}{2} < x < \frac{{9\pi }}{2}$

Note: If working shown, award A1A0 if inclusion/exclusion of endpoints is incorrect. If no working shown award N1.
If working shown, award A1A0 if both correct intervals are given, and additional intervals are given. If no working shown award N1.
Award A0A0 if inclusion/exclusion of endpoints are incorrect and additional intervals are given.

[8 marks]

Examiners report

[N/A]

Syllabus sections

Topic 3— Geometry and trigonometry » SL 3.8—Solving trig equations

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