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Date	May Specimen paper	Marks available	5	Reference code	SPM.1.SL.TZ0.6
Level	Standard Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 0
Command term	Solve and Hence or otherwise	Question number	6	Adapted from	N/A

Question

Show that ${\text{lo}}{{\text{g}}_9}\left( {{\text{cos}}\,2x + 2} \right) = {\text{lo}}{{\text{g}}_3}\sqrt {{\text{cos}}\,2x + 2}$ .

[3]

a.

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 $0 < x < \frac{\pi }{2}$ .

[5]

b.

Markscheme

attempting to use the change of base rule M1

${\text{lo}}{{\text{g}}_9}\left( {{\text{cos}}\,2x + 2} \right) = \frac{{{\text{lo}}{{\text{g}}_3}\left( {{\text{cos}}\,2x + 2} \right)}}{{{\text{lo}}{{\text{g}}_3}9}}$ A1

$= \frac{1}{2}{\text{lo}}{{\text{g}}_3}\left( {{\text{cos}}\,2x + 2} \right)$ A1

$= {\text{lo}}{{\text{g}}_3}\sqrt {{\text{cos}}\,2x + 2}$ AG

[3 marks]

a.

${\text{lo}}{{\text{g}}_3}\left( {{\text{2}}\,{\text{sin}}\,x} \right) = {\text{lo}}{{\text{g}}_3}\sqrt {{\text{cos}}\,2x + 2}$

${\text{2}}\,{\text{sin}}\,x = \sqrt {{\text{cos}}\,2x + 2}$ M1

${\text{4}}\,{\text{si}}{{\text{n}}^2}\,x = {\text{cos}}\,2x + 2$ (or equivalent) A1

use of ${\text{cos}}\,2x = 1 - 2\,{\text{si}}{{\text{n}}^2}\,x$ (M1)

$6\,{\text{si}}{{\text{n}}^2}\,x = 3$

${\text{sin}}\,x = \left( \pm \right)\frac{1}{{\sqrt 2 }}$ A1

$x = \frac{\pi }{4}$ A1

Note: Award A0 if solutions other than $x = \frac{\pi }{4}$ are included.

[5 marks]

b.

Examiners report

[N/A]

a.

[N/A]

b.

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 $\cos \theta$ .
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:
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20N.1.SL.TZ0.S_2b:
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SPM.1.SL.TZ0.1b:
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21M.1.SL.TZ1.6a:
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21M.1.SL.TZ2.3b:
Hence, solve the equation $2 \cos^{2} x + 5 \sin x = 4, 0 \leq x \leq 2 π$ .
17M.1.SL.TZ2.S_7:
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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:
Find $\cos 2\theta$ .
17M.1.SL.TZ1.S_10a:
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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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