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Question

 Let $$f(x)= \sin ^{23} x-\cos ^{22} x$$ and $$ \displaystyle g(x)=1+\frac{1}{2} \tan^{-1}\left|x\right|.$$
Then, the number of values of $$ x$$ in interval $$[-10\pi,20\pi]$$ satisfying  the equation $$ f(x)= sgn(g(x)),$$ is $$5a$$ Then, $$a$$ is


Solution

$$ \displaystyle g(x)=\frac{1}{2} \tan^{-1} \left|x\right| +1$$
$$ \Rightarrow sgn(g(x))=1$$
$$\sin^{23}x-\cos ^{22} x=1$$
$$ \sin^{23}x=1+\cos ^{22}x$$
which is possible ,if $$\sin x=1$$ and $$ \displaystyle \cos x=0 \Rightarrow \sin x=1,x=2n\pi +\frac{\pi}{2}$$
Hence, $$ \displaystyle -10\pi \leq 2n\pi +\frac{\pi}{2}\leq 20\pi $$
$$\displaystyle \Rightarrow -\frac{21}{2}\leq 2n\leq \frac{39}{2}$$
$$\displaystyle \Rightarrow -\frac{21}{4}\leq n\leq \frac{39}{4}\Rightarrow -5 \leq n\leq 9 $$
Hence, number of values of $$ x=15\Rightarrow a=3$$
Ans: 3

Mathematics

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