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Date May 2014 Marks available 2 Reference code 14M.1.hl.TZ1.5
Level HL only Paper 1 Time zone TZ1
Command term Prove that Question number 5 Adapted from N/A

Question

Use the identity cos2θ=2cos2θ1 to prove that cos12x=1+cosx2, 0.

[2]
a.

Find a similar expression for \sin \frac{1}{2}x,{\text{ }}0 \leqslant x \leqslant \pi .

[2]
b.

Hence find the value of \int_0^{\frac{\pi }{2}} {\left( {\sqrt {1 + \cos x}  + \sqrt {1 - \cos x} } \right){\text{d}}x} .

[4]
c.

Markscheme

\cos x = 2{\cos ^2}\frac{1}{2}x - 1

\cos \frac{1}{2}x =  \pm \sqrt {\frac{{1 + \cos x}}{2}}     M1

positive as 0 \leqslant x \leqslant \pi     R1

\cos \frac{1}{2}x = \sqrt {\frac{{1 + \cos x}}{2}}     AG

[2 marks]

a.

\cos 2\theta  = 1 - 2{\sin ^2}\theta     (M1)

\sin \frac{1}{2}x = \sqrt {\frac{{1 - \cos x}}{2}}     A1

[2 marks]

b.

\sqrt 2 \int_0^{\frac{\pi }{2}} {\cos \frac{1}{2}x + \sin \frac{1}{2}x{\text{d}}x}     A1

= \sqrt 2 \left[ {2\sin \frac{1}{2}x - 2\cos \frac{1}{2}x} \right]_0^{\frac{\pi }{2}}     A1

= \sqrt 2 (0) - \sqrt 2 (0 - 2)     A1

= 2\sqrt 2     A1

[4 marks]

c.

Examiners report

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a.
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b.
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c.

Syllabus sections

Topic 3 - Core: Circular functions and trigonometry » 3.3 » Double angle identities.

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