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Date	May 2022	Marks available	5	Reference code	22M.1.SL.TZ1.4
Level	Standard Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 1
Command term	Solve	Question number	4	Adapted from	N/A

Question

Consider the functions $f (x) = \sqrt{3} \sin x + \cos x$ where $0 \leq x \leq π$ and $g (x) = 2 x$ where $x \in ℝ$ .

Find $(f \circ g) (x)$ .

[2]

Solve the equation $(f \circ g) (x) = 2 \cos 2 x$ where $0 \leq x \leq π$ .

[5]

Markscheme

$(f \circ g) (x) = f (2 x)$ (A1)

$f (2 x) = \sqrt{3} \sin 2 x + \cos 2 x$ A1

[2 marks]

$\sqrt{3} \sin 2 x + \cos 2 x = 2 \cos 2 x$

$\sqrt{3} \sin 2 x = \cos 2 x$

recognising to use $\tan$ or $cot$ M1

$\tan 2 x = \frac{1}{\sqrt{3}}$ OR $cot 2 x = \sqrt{3}$ (values may be seen in right triangle) (A1)

$(arctan (\frac{1}{\sqrt{3}}) =) \frac{π}{6}$ (seen anywhere) (accept degrees) (A1)

$2 x = \frac{π}{6}, \frac{7 π}{6}$

$x = \frac{π}{12}, \frac{7 π}{12}$ A1A1

Note: Do not award the final A1 if any additional solutions are seen.
Award A1A0 for correct answers in degrees.
Award A0A0 for correct answers in degrees with additional values.

[5 marks]

Examiners report

Determining the composite function was very well done. In part (b) very few candidates showed any recognition that tan (or cot) were required to solve this trigonometric equation. Many saw the 2x and simply employed one of the double angle rules but could not then progress to an answer.

[N/A]

Syllabus sections

Topic 3— Geometry and trigonometry » SL 3.5—Unit circle definitions of sin, cos, tan. Exact trig ratios, ambiguous case of sine rule

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19M.2.SL.TZ2.S_3a:
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19M.2.SL.TZ2.S_3b:
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19M.1.SL.TZ2.S_9a:
Find the value of ${\text{tan}}\,\theta$ .
22M.1.SL.TZ1.9b.i:
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22M.1.SL.TZ1.9b.ii:
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22M.1.AHL.TZ1.9b:
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22M.1.AHL.TZ2.7:
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18M.2.SL.TZ1.S_6:
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19N.1.AHL.TZ0.H_9d:
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22M.1.SL.TZ2.5:
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22M.2.AHL.TZ2.7a:
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19N.1.AHL.TZ0.H_9c.ii:
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17N.1.SL.TZ0.S_4a:
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16N.2.AHL.TZ0.H_7a:
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16N.1.AHL.TZ0.H_13a:
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17M.1.SL.TZ1.S_3:
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