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${\rm{Simplify, }}\sqrt {220 - 30\sqrt {35} } $

The given expression, 

${ = \sqrt {220 - 2 \times 15 \times \sqrt {35} } }$

${ = \sqrt {220 - 2 \times 3 \times 5 \times \sqrt 7  \times \sqrt 5 } }$

${ = \sqrt {\underbrace {{{(3\sqrt 5 )}^2}}_{9 \times 5 = 45} + \underbrace {{{(5\sqrt 7 )}^2}}_{25 \times 7 = 175} - 2(3\sqrt 5 )(5\sqrt 7 )} }$

${ = \sqrt {{{(3\sqrt 5  - 5\sqrt 7 )}^2}} }$

${ = |3\sqrt 5  - 5\sqrt 7 | = 5\sqrt 7  - 3\sqrt 5 }$

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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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