Q235 / 270 · NDA 2019 (Phase 2) · Mathematics

Paragraph: Consider the equation $x^y=e^{x-y}$
Question: What is $\frac{d^2 y}{d x^2}$ at $x=1$ equal to?

  • A
    0
  • Correct answer
    B
    1
  • C
    2
  • D
    4

Solution

$\begin{aligned} & \text { } \because y^1=\frac{x-y}{x(1+\ell \mathrm{n} x)} \\ & \Rightarrow y^{11}=\frac{x(1+\ell \mathrm{n} x)\left(1-y^1\right)-(x-y)(1+1+\ell \mathrm{n} x)}{x^2(1+\ell \mathrm{n} x)^2} \\ & \Rightarrow y^{11} \text { at } x=1 \\ & \Rightarrow y^{11}=\frac{1(1)(1)-0}{1(1)} \\ & \Rightarrow y^{11}=1\end{aligned}$

Hence, the correct answer is option: 'B'.

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