Question

Total number of positive integral value of `n' such that the equations $co{s}^{-1}x+(si{n}^{-1}y{)}^2=\frac{n{\pi }^2}{4}$ and $(si{n}^{-1}y{)}^2-co{s}^{-1}x=\frac{{\pi }^2}{16}$ are consistent, is equal to :

• A. 1
• B. 4
• C. 3
• D. 2

Answer 1

The correct option is A 1

$(si{n}^{-1}y{)}^2+co{s}^{-1}x=\frac{n{\pi }^2}{4}$
$(si{n}^{-1}y{)}^2-co{s}^{-1}x=\frac{{\pi }^2}{16}$
$\Rightarrow (si{n}^{-1}y{)}^2=\frac{(4n+1){\pi }^2}{32},co{s}^{-1}x=\frac{{\pi }^2(4n-1)}{32}$
$\Rightarrow 0\le \frac{(4n+1)}{32}{\pi }^2\le \frac{{\pi }^2}{4},0\le \frac{(4n-1)}{32}{\pi }^2\le \pi$
$\Rightarrow -\frac{1}{4}\le n\le \frac{7}{4},\frac{1}{4},\le n\le \frac{8}{\pi }+\frac{1}{4}$