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STD 12 - 1. relation and function — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 12 - 1. relation and function4 Marks
Question
Let $f: R \rightarrow R$ be a function defined $f(x)=\frac{2 e^{2 x}}{e^{2 x}+\varepsilon}$. Then $f\left(\frac{1}{100}\right)+f\left(\frac{2}{100}\right)+f\left(\frac{3}{100}\right)+\ldots .+f\left(\frac{99}{100}\right)$ is equal to
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Answer

b
$f(x)+f(1-x)=\frac{2 e^{2 x}}{e^{2 x}+e}+\frac{2 e^{2-2 x}}{e^{2-e x}+e}=\left[\frac{e^{2 x}}{e^{2 x}+e}+\frac{e^{2}}{e^{2}+e^{2 x+1}}\right]$

$=2\left[\frac{e^{2 x-1}}{e^{2 x-1}+1}+\frac{1}{1+e^{2 x-1}}\right]=2$

$f\left(\frac{1}{100}\right)+f\left(\frac{2}{100}\right)+f\left(\frac{3}{100}\right)+\ldots .+f\left(\frac{99}{100}\right)$

$=\left\{f\left(\frac{1}{100}\right)+f\left(\frac{99}{100}\right)\right\}+\left\{f\left(\frac{2}{100}\right)+f\left(\frac{98}{100}\right)\right\}+\ldots .+f\left\{\left(\frac{49}{100}\right)+f\left(\frac{51}{100}\right)\right\}+f\left(\frac{1}{2}\right)$

$=(2+2+2+----49$ times $)+\frac{2 e}{e+e}$

$=98+1=99$

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