Question

For \(y=\sin ^{-1}\left\{\frac{5 x+12 \sqrt{1-x^{2}}}{13}\right\} ;|x| \leq 1\), if \(a\left(1-x^{2}\right) y_{2}+b x y_{1}=0\) then \((a, b)=\)
(A) \((2,1)\)
(B) \((1,-1)\)
(C) \((-1,1)\)
(D) \((1,2)\)

Answer 1

Ans: (B)
Hint \(: y=\sin ^{-1}\left(\frac{5 x+12 \sqrt{1-x^{2}}}{13}\right)\)
Let \(\sin \theta_{1}=\frac{5}{13}\) and \(\cos \theta_{2}=\mathrm{x}\)
$$
\begin{aligned}
&y=\sin ^{-1}\left(\sin \theta_{1} \cdot \cos \theta_{2}+\cos \theta_{1} \cdot \sin \theta_{2}\right) \\
&y=\theta_{1}+\theta_{2}=\sin ^{-1} \frac{5}{13}+\cos ^{-1} x
\end{aligned}
$$
$$
\begin{aligned}
&y_{1}=-\frac{1}{\sqrt{1-x^{2}}} \\
&y_{2}=\frac{-x}{\left(1-x^{2}\right) \sqrt{1-x^{2}}} \\
&\Rightarrow y_{2}\left(1-x^{2}\right)=x \cdot y_{1} \\
&\Rightarrow y_{2}\left(1-x^{2}\right)-x y_{1}=0 \\
&\therefore a=1, b=-1 \\
&(a, b)=(1,-1)
\end{aligned}
$$

Answer 2

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Answer 3

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