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UPSC NDA 2 2019 Mathematics Solved Paper

Directions (Q. Nos. 84 and 85): Read the following information and answer the two items that follow:

Consider the equation

x^y=e^{x–y}

[NDA II 2019]

Question : 85 of 120

Marks: +1, -0

What is

\frac{d^2y}{dx^2}

Solution: 👈: Video Solution

\frac{dy}{dx} = \frac{\log x}{(1+\log x)^2}

from (ii)

\Rightarrow \frac{d^2y}{dx^2} = \frac{ (1\log x)^2 \cdot \frac{1}{x} - \log x \cdot 2(1+\log x) \cdot \left(0 + \frac{1}{x}\right) }{ (1+\log x)^2 }

= \frac{ (1+\log x)^2 \cdot \frac{1}{x} - \frac{2}{x} \log x (1+\log x) }{ (1+\log x)^2 }

\therefore \left( \frac{d^2y}{dx^2} \right)_{x=1} = \frac{ (1+\log 1)^2 \cdot \frac{1}{1} - \frac{2}{1} \log 1 (1+\log 1) }{ (1+\log 1)^2 }

= \frac{1-2\cdot0}{(1+0)^2} = \frac{1-0}{1} = 1

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