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1.Relation and Function
hard
माना $f: R \rightarrow R$ एक फलन है, जो $f(x)=\frac{2 e^{2 x}}{e^{2 x}+e}$ तब $f\left(\frac{1}{100}\right)+f\left(\frac{2}{100}\right)+f\left(\frac{3}{100}\right)+\ldots . .+f\left(\frac{99}{100}\right)$ बराबर होगा।
A$98$
B$99$
C$100$
D$101$
(JEE MAIN-2022)
Solution
$f(x)+f(1-x)=\frac{2 e^{2 x}}{e^{2 x}+e}+\frac{2 e^{2-2 x}}{e^{2-e x}+e}=\left[\frac{e^{2 x}}{e^{2 x}+e}+\frac{e^{2}}{e^{2}+e^{2 x+1}}\right]$
$=2\left[\frac{e^{2 x-1}}{e^{2 x-1}+1}+\frac{1}{1+e^{2 x-1}}\right]=2$
$f\left(\frac{1}{100}\right)+f\left(\frac{2}{100}\right)+f\left(\frac{3}{100}\right)+\ldots .+f\left(\frac{99}{100}\right)$
$=\left\{f\left(\frac{1}{100}\right)+f\left(\frac{99}{100}\right)\right\}+\left\{f\left(\frac{2}{100}\right)+f\left(\frac{98}{100}\right)\right\}+\ldots .+f\left\{\left(\frac{49}{100}\right)+f\left(\frac{51}{100}\right)\right\}+f\left(\frac{1}{2}\right)$
$=(2+2+2+—-49$ times $)+\frac{2 e}{e+e}$
$=98+1=99$
$=2\left[\frac{e^{2 x-1}}{e^{2 x-1}+1}+\frac{1}{1+e^{2 x-1}}\right]=2$
$f\left(\frac{1}{100}\right)+f\left(\frac{2}{100}\right)+f\left(\frac{3}{100}\right)+\ldots .+f\left(\frac{99}{100}\right)$
$=\left\{f\left(\frac{1}{100}\right)+f\left(\frac{99}{100}\right)\right\}+\left\{f\left(\frac{2}{100}\right)+f\left(\frac{98}{100}\right)\right\}+\ldots .+f\left\{\left(\frac{49}{100}\right)+f\left(\frac{51}{100}\right)\right\}+f\left(\frac{1}{2}\right)$
$=(2+2+2+—-49$ times $)+\frac{2 e}{e+e}$
$=98+1=99$
Standard 12
Mathematics
