19N.1.SL.TZ0.S_6

pestleMathematicsAASLPaper 119N· sl-3-8-solving-trig-equationssource ↗

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 / solution

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)

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)

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

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

[8 marks]

Examiners’ report

[N/A]