JEE Mains · Maths · STD 12 - 5. continuity and differentiation
Let \(f(x) = log_e\,(sin\,x),\) \((0\,<\,x\,< \pi )\) and \(g(x) = sin^{-1}\,(e^{-x}),\) \((x\, \ge \,0)\). If \(\alpha \) is a positive real number such that \(a\) \( = (fog)’(\alpha )\) and \(b = (fog)(\alpha ),\) then
- A \(a{x^2}\, + \,b\alpha \, - a\, = 2{\alpha ^2}\)
- B \(a{x^2}\, - \,b\alpha \, - a\, = 0\)
- C \(a{x^2}\, - \,b\alpha \, - a\, = 1\)
- D \(a{x^2}\, + \,b\alpha \, + a\, = 0\)
Answer & Solution
Correct Answer
(C) \(a{x^2}\, - \,b\alpha \, - a\, = 1\)
Step-by-step Solution
Detailed explanation
\(fog\,(x)\, = \,( - x)\, \Rightarrow \,\left( {fog\left( \alpha \right)} \right)\, = \, - \,\alpha \, = \,b\) \((fog\,(x))'\, = \, - 1\, \Rightarrow \,\left( {fog\left( \alpha \right)} \right)'\, = \, - \,1\, = \,a\)
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