$ f^2(x)=\operatorname{Lim}_{r \rightarrow x}\left(\frac{2 r^2\left(f^2(r)-f(x) f(r)\right)}{r^2-x^2}-r^3 e^{\frac{f(r)}{r}}\right) $
$ =\operatorname{Lim}_{r \rightarrow x}\left(\frac{2 r^2 f(r)}{r+x} \frac{(f(r)-f(x))}{r-x}-r^3 e^{\frac{f(r)}{r}}\right) $
$ f^2(x)=\frac{2 x^2 f(x)}{2 x} f^{\prime}(x)-x^3 e^{\frac{f(x)}{x}} $
$ y^2=x y \frac{d y}{d x}-x^3 e^{\frac{y}{x}} $
$ \frac{y}{x}=\frac{d y}{d x}-\frac{x^2}{y} e^{\frac{y}{x}}$
Put $y=v x \Rightarrow \frac{d y}{d x}=v+x \frac{d v}{d x}$
$ v=v+x \frac{d v}{d x}-\frac{x}{v} e^{-} $
$ \frac{d v}{d x}=\frac{e^r}{v} \Rightarrow e^{-r} v d v=d x$
Integrating both side
$ \mathrm{e}^{\mathrm{v}}(\mathrm{x}+\mathrm{c})+1+\mathrm{v}=0$
$ \mathrm{f}(\mathrm{l})=1 \Rightarrow \mathrm{x}=1, \mathrm{y}=1$
$\Rightarrow c=-1-\frac{2}{e} $
$ e^{-}\left(-1-\frac{2}{e}+x\right)+1+v=0 $
$ e^{\frac{y}{x}}\left(-1-\frac{2}{e}+x\right)+1+\frac{y}{x}=0 $
$ x=a, y=0 \Rightarrow a=\frac{2}{e} $
$ a e=2$
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
| $Face:$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ |
| $P(F)$ | $0.1$ | $0.24$ | $0.19$ | $0.18$ | $0.15$ | $0.14$ |
If an even face has turned up, then the probability that it is face $2$ or face $4$, is