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1.Relation and Function
hard
The largest interval lying in $\left( { - \frac{\pi }{2},\frac{\pi }{2}} \right)$ for which the function, $f\left( x \right) = {4^{ - {x^2}}} + {\cos ^{ - 1}}\left( {\frac{x}{2} - 1} \right) + \log \left( {\cos x} \right)$ is defined is
A
$\left[ { - \frac{\pi }{4},\frac{\pi }{2}} \right)$
B
$\left[ {0,\frac{\pi }{2}} \right)$
C
$\left[ {0,\pi } \right]$
D
$\;\left( { - \frac{\pi }{2},\frac{\pi }{2}} \right)$
(AIEEE-2007)
Solution
$F(x)=4^{-x^{2}}+\cos ^{-1}\left(\frac{x}{2}-1\right)+\log (\cos x)$
$-1 \leq \frac{x}{2}-1 \leq 1$
$0 \leq \frac{x}{2} \leq 2$
$0 \leq x \leq 4 \ldots \ldots .(1)$
$\log (\cos x) w h e n \cos x>0$
$\cos >0$
$x=\left(0, \frac{\pi}{2}\right) \dots(2)$
final $x=\left(0, \frac{\pi}{2}\right)$
Standard 12
Mathematics
