12th Class Mathematics Sample Paper Mathematics Olympiad - Sample Paper-1

  • question_answer
    The solution of \[\frac{dy}{dx}+\sqrt{\frac{1-{{y}^{2}}}{1+{{x}^{2}}}}=0\] is

    A)  \[si{{n}^{-1}}x.si{{n}^{-1}}y=c\]     

    B)  \[si{{n}^{-1}}x-si{{n}^{-1}}y=c\]

    C)  \[si{{n}^{-1}}x+si{{n}^{-1}}y=c\]   

    D)  \[si{{n}^{-1}}x=c.si{{n}^{-1}}y\]

    Correct Answer: C

    Solution :

    [c]  \[\because \frac{dx}{dy}+\sqrt{\frac{1-{{y}^{2}}}{1-{{x}^{2}}}}=0\] \[\Rightarrow \sqrt{\frac{1-{{y}^{2}}}{1-{{x}^{2}}}}=-\frac{dy}{dx}\] \[\Rightarrow \frac{1}{\sqrt{1-{{x}^{2}}}}.dx=-\frac{1}{1-{{y}^{2}}}.dy\] Integrating both sides, we have \[\int{\frac{1}{\sqrt{1-{{x}^{2}}}}.dx}+\int{\frac{1}{\sqrt{1-{{y}^{2}}}}.dy=0\Rightarrow {{\sin }^{-1}}x}+{{\sin }^{-1}}y=c\]which is the required solution. Hence, option [c] is correct.


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