Inverse Trigonometric Functions Question 4
Question: Total number of positive integral value ?n? so that the equations $ {{\cos }^{-1}}x+{{(si{n^{-1}}y)}^{2}}=\frac{n{{\pi }^{2}}}{4} $ and $ {{(si{n^{-1}}y)}^{2}}-{{\cos }^{-1}}x=\frac{{{\pi }^{2}}}{16} $ are consistent, is equal to
Options:
A) 1
B) 4
C) 3
D) 2
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Answer:
Correct Answer: A
We have, $ 2{{(si{n^{-1}}y)}^{2}}=\frac{4n+1}{16}{{\pi }^{2}} $
$ \Rightarrow 0\le \frac{4n+1}{32}{{\pi }^{2}}\le \frac{{{\pi }^{2}}}{4} $
Also, $ 2(co{s^{-1}}x)=\frac{4n-1}{16}{{\pi }^{2}}\Rightarrow -\frac{1}{4}\le n\le \frac{7}{4} $
Also, $ 2(co{s^{-1}}x)=\frac{4n-1}{16}{{\pi }^{2}} $
$ \Rightarrow 0\le \frac{4n-1}{32}{{\pi }^{2}}\le \pi \Rightarrow \frac{1}{4}\le n\le \frac{8}{\pi }+\frac{1}{4}\Rightarrow n=1 $
BETA
