JEE Mains · Maths · STD 12 - 5. continuity and differentiation
Let \(\mathrm{f}(\mathrm{x})=\cos \left(2 \tan ^{-1} \sin \left(\cot ^{-1} \sqrt{\frac{1-\mathrm{x}}{\mathrm{x}}}\right)\right)\) \(0<\mathrm{x}<1\). Then :
- A \((1-x)^{2} f^{\prime}(x)-2(f(x))^{2}=0\)
- B \((1+x)^{2} f^{\prime}(x)+2(f(x))^{2}=0\)
- C \((1-x)^{2} f^{\prime}(x)+2(f(x))^{2}=0\)
- D \((1+x)^{2} f^{\prime}(x)-2(f(x))^{2}=0\)
Answer & Solution
Correct Answer
(C) \((1-x)^{2} f^{\prime}(x)+2(f(x))^{2}=0\)
Step-by-step Solution
Detailed explanation
\(f(x)=\cos \left(2 \tan ^{-1} \sin \left(\cot ^{-1} \sqrt{\frac{1-x}{x}}\right)\right)\) \(\cot ^{-1} \sqrt{\frac{1-x}{x}}=\sin ^{-1} \sqrt{x}\) \(\text { or } f(x)=\cos \left(2 \tan ^{-1} \sqrt{x}\right)\) \(=\cos \tan ^{-1}\left(\frac{2 \sqrt{x}}{1-x}\right)\)…
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