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Simplify, $\frac{{\sqrt {15} + \sqrt {35} + \sqrt {21} + 5}}{{\sqrt 3 + 2\sqrt 5 + \sqrt 7 }}$

 The given expression can be rearranged as $ \frac{{(\sqrt {15}  + 5) + (\sqrt {35}  + \sqrt {21} )}}{{\sqrt 3  + 2\sqrt 5  + \sqrt 7 }}$

$ = \frac{{\sqrt 5 (\sqrt 3  + \sqrt 5 ) + \sqrt 7 (\sqrt 5  + \sqrt 3 )}}{{\sqrt 3  + 2\sqrt 5  + \sqrt 7 }}$

$ = \frac{{(\sqrt 3  + \sqrt 5 )(\sqrt 5  + \sqrt 7 )}}{{\sqrt 3  + 2\sqrt 5  + \sqrt 7 }}= \frac{{(\sqrt 3  + \sqrt 5 )(\sqrt 5  + \sqrt 7 )}}{{(\sqrt 3  + \sqrt 5 ) + (\sqrt 5  + \sqrt 7 )}}$

$ = \frac{1}{{\frac{{(\sqrt 3  + \sqrt 5 )}}{{(\sqrt 3  + \sqrt 5 )(\sqrt 5  + \sqrt 7 )}} + \frac{{(\sqrt 5  + \sqrt 7 )}}{{(\sqrt 3  + \sqrt 5 )(\sqrt 5  + \sqrt 7 )}}}}$

$ = \frac{1}{{\frac{1}{{(\sqrt 5  + \sqrt 7 )}} + \frac{1}{{(\sqrt 3  + \sqrt 5 )}}}}$

$ = \frac{1}{{\frac{{(\sqrt 7  - \sqrt 5 )}}{{(\sqrt 7  + \sqrt 5 )(\sqrt 7  - \sqrt 5 )}} + \frac{{(\sqrt 5  - \sqrt 3 )}}{{(\sqrt 5  + \sqrt 3 )(\sqrt 5  - \sqrt 3 )}}}}$

$ = \frac{1}{{\frac{{(\sqrt 7  - \sqrt 5 )}}{2} + \frac{{(\sqrt 5  - \sqrt 3 )}}{2}}}$

$ = \frac{2}{{(\sqrt 7  - \sqrt 3 )}} = \frac{{2(\sqrt 7  + \sqrt 3 )}}{{(\sqrt 7  - \sqrt 3 )(\sqrt 7  + \sqrt 3 )}} = \frac{{\sqrt 7  + \sqrt 3}}{2}$

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${\log _{\sqrt 5 }}\left[ {3 + \cos \left( {\frac{{3\pi }}{4} + x} \right) + \cos \left( {\frac{\pi }{4} + x} \right) + \cos \left( {\frac{\pi }{4} - x} \right) - \cos \left( {\frac{{3\pi }}{4} - x} \right)} \right]$

The range of the function $f(x) = {\log _{\sqrt 5 }}\left[ {3 + \cos \left( {\frac{{3\pi }}{4} + x} \right) + \cos \left( {\frac{\pi }{4} + x} \right) + \cos \left( {\frac{\pi }{4} - x} \right) - \cos \left( {\frac{{3\pi }}{4} - x} \right)} \right]$ is: (A) $[ - 2,2]$ (B) $\left[ {\frac{1}{{\sqrt 5 }},\sqrt 5 } \right]$ (C) $(0,\sqrt 5 )$ (D) $[ 0,2]$ Solution We have, $f(x) = {\log _{\sqrt 5 }}\left( {3 - 2\sin \frac{{3\pi }}{4}\sin x + 2\cos \frac{\pi }{4}\cos x} \right)$ $ \Rightarrow f(x) = {\log _{\sqrt 5 }}\left[ {3 + \sqrt 2 (\cos x - \sin x)} \right]$ Now, $ - \sqrt 2  \le \cos x - \sin x \le \sqrt 2 $ $\therefore - 2 \le \sqrt 2 (\cos x - \sin x) \le 2$ $\therefore 1 \le 3 + \sqrt 2 (\cos x - \sin x) \le 5$ $\therefore{\log _{\sqrt 5 }}1 \le {\log _{\sqrt 5 }}[3 + \sqrt 2 (\cos x - \sin x)] \le {\log _{\sqrt 5 }}5$ $ \Rightarrow 0 \le f(x) \le 2$ Answer: (D)
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