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Date	May 2017	Marks available	7	Reference code	17M.1.SL.TZ2.S_7
Level	Standard Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 2
Command term	Solve	Question number	S_7	Adapted from	N/A

Question

Solve ${\log _2}(2\sin x) + {\log _2}(\cos x) = - 1$ , for $2\pi < x < \frac{{5\pi }}{2}$ .

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

correct application of $\log a + \log b = \log ab$ (A1)

eg $\,\,\,\,\,$ ${\log _2}(2\sin x\cos x),{\text{ }}\log 2 + \log (\sin x) + \log (\cos x)$

correct equation without logs A1

eg $\,\,\,\,\,$ $2\sin x\cos x = {2^{ - 1}},{\text{ }}\sin x\cos x = \frac{1}{4},{\text{ }}\sin 2x = \frac{1}{2}$

recognizing double-angle identity (seen anywhere) A1

eg $\,\,\,\,\,$ $\log (\sin 2x),{\text{ }}2\sin x\cos x = \sin 2x,{\text{ }}\sin 2x = \frac{1}{2}$

evaluating ${\sin ^{ - 1}}\left( {\frac{1}{2}} \right) = \frac{\pi }{6}{\text{ }}(30^\circ )$ (A1)

correct working A1

eg $\,\,\,\,\,$ $x = \frac{\pi }{{12}} + 2\pi ,{\text{ }}2x = \frac{{25\pi }}{6},{\text{ }}\frac{{29\pi }}{6},{\text{ }}750^\circ ,{\text{ }}870^\circ ,{\text{ }}x = \frac{\pi }{{12}}$ and $x = \frac{{5\pi }}{{12}}$ , one correct final answer

$x = \frac{{25\pi }}{{12}},{\text{ }}\frac{{29\pi }}{{12}}$ (do not accept additional values) A2 N0

[7 marks]

Examiners report

[N/A]

Syllabus sections

Topic 3— Geometry and trigonometry » SL 3.6—Pythagorean identity, double angles

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Use the principle of mathematical induction to prove that

$\sin x + \sin 3x + \ldots + \sin (2n - 1)x = \frac{{1 - \cos 2nx}}{{2\sin x}},{\text{ }}n \in {\mathbb{Z}^ + },{\text{ }}x \ne k\pi$ where $k \in \mathbb{Z}$ .
16N.1.SL.TZ0.S_2a:
Find $\cos \theta$ .
SPM.1.SL.TZ0.6b:
Hence or otherwise solve ${\text{lo}}{{\text{g}}_3}\left( {{\text{2}}\,{\text{sin}}\,x} \right) = {\text{lo}}{{\text{g}}_9}\left( {{\text{cos}}\,2x + 2} \right)$ for $0 < x < \frac{\pi }{2}$ .
16N.1.AHL.TZ0.H_13b:
Show that $\frac{{1 - \cos 2x}}{{2\sin x}} \equiv \sin x,{\text{ }}x \ne k\pi$ where $k \in \mathbb{Z}$ .
17M.1.SL.TZ1.S_10b:
Given that $\tan \theta > 0$ , find $\tan \theta$ .
SPM.1.SL.TZ0.1a:
Given that ${\text{cos}}\,\hat A = \frac{5}{6}$ find the value of ${\text{sin}}\,\hat A$ .
19M.1.SL.TZ2.S_9a:
Find the value of ${\text{tan}}\,\theta$ .
17M.1.SL.TZ1.S_10c:
Let $y = \frac{1}{{\cos x}}$ , for $0 < x < \frac{\pi }{2}$ . The graph of $y$ between $x = \theta$ and $x = \frac{\pi }{4}$ is rotated 360° about the $x$ -axis. Find the volume of the solid formed.
SPM.1.SL.TZ0.6a:
Show that ${\text{lo}}{{\text{g}}_9}\left( {{\text{cos}}\,2x + 2} \right) = {\text{lo}}{{\text{g}}_3}\sqrt {{\text{cos}}\,2x + 2}$ .
22M.1.SL.TZ1.4b:
Solve the equation $(f \circ g) (x) = 2 \cos 2 x$ where $0 \leq x \leq π$ .
22M.1.SL.TZ1.9a.ii:
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20N.1.SL.TZ0.S_2a:
Given that $A \hat{B} C$ is acute, find $\sin θ$ .
20N.1.SL.TZ0.S_2b:
Find $\cos (2 \times C \hat{A} B)$ .
SPM.1.SL.TZ0.1b:
Find the area of triangle ABC.
21M.1.SL.TZ1.6a:
Show that $\sin 2 x + \cos 2 x - 1 = 2 \sin x (\cos x - \sin x)$ .
21M.1.SL.TZ2.3b:
Hence, solve the equation $2 \cos^{2} x + 5 \sin x = 4, 0 \leq x \leq 2 π$ .
18N.1.SL.TZ0.S_7:
Given that ${\text{sin}}\,x = \frac{1}{3}$ , where $0 < x < \frac{\pi }{2}$ , find the value of ${\text{cos}}\,4x$ .
16N.1.AHL.TZ0.H_13d:
Hence or otherwise solve the equation $\sin x + \sin 3x = \cos x$ in the interval $0 < x < \pi$ .
21M.1.AHL.TZ1.12a:
Show that $f'' (x) = - \frac{1}{4 \sqrt{{(1 + x)}^{3}}}$ .
16N.1.SL.TZ0.S_2b:
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17M.1.SL.TZ1.S_10a:
Show that $\cos \theta = \frac{3}{4}$ .
21M.1.SL.TZ1.6b:
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21M.1.SL.TZ2.3a:
Show that the equation $2 \cos^{2} x + 5 \sin x = 4$ may be written in the form $2 \sin^{2} x - 5 \sin x + 2 = 0$ .

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