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  • Provide solution for RD Sharma maths class 12 chapter 10 Differentiation exercise Multiple choice question 27

Provide solution for RD Sharma maths class 12 chapter 10 Differentiation exercise Multiple choice question 27

Answers (1)

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Answer:

        \frac{x^{2}}{y^{2}} \sqrt{\frac{1-y^{6}}{1-x^{6}}}

Hint:

        Differentiate the function w.r.t x

Given:

        \sqrt{1-x^{6}}+\sqrt{1-y^{6}}=a^{3}\left(x^{3}-y^{3}\right)

Solution:  

We have, \sqrt{1-x^{6}}+\sqrt{1-y^{6}}=a^{3}\left(x^{3}-y^{3}\right)

Putting x^{3}=\sin A, y^{3}=\sin B

        \begin{aligned} &\sqrt{1-\sin ^{2} A}+\sqrt{1-\sin ^{2} B}=a(\sin A-\sin B) \\\\ &\cos A+\cos B=a(\sin A-\sin B) \end{aligned}

        \begin{aligned} &2 \cos \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)=2 a \sin \left(\frac{A-B}{2}\right) \cos \left(\frac{A+B}{2}\right) \\\\ &\cot \left(\frac{A-B}{2}\right)=a^{3} \end{aligned}

        \begin{aligned} &\frac{A-B}{2}=\cot ^{-1}\left(a^{3}\right) \\\\ &A-B=2 \cot ^{-1}\left(a^{3}\right) \\\\ &\sin ^{-1} x^{3}-\sin ^{-1} y^{3}=2 \cot ^{-1}\left(a^{3}\right) \end{aligned}

        \begin{aligned} &\frac{1}{\sqrt{1-x^{6}}} \times \frac{d}{d x}\left(x^{3}\right)-\frac{1}{\sqrt{1-y^{6}}} \times \frac{d}{d x}\left(y^{3}\right)=0 \\\\ &\frac{1}{\sqrt{1-x^{6}}} \times 3 x^{2}-\frac{1}{\sqrt{1-y^{6}}} \times 3 y^{2} \times \frac{d y}{d x}=0 \end{aligned}

        \frac{d y}{d x}=\frac{x^{2}}{y^{2}} \sqrt{\frac{1-y^{6}}{1-x^{6}}}

 

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