MCQ
$\alpha$ ની કઈ કિમત માટે $4 \alpha \int\limits_{-1}^{2} \mathrm{e}^{-\alpha \mathrm{|x|} } \mathrm{d} \mathrm{x}=5 $ થાય .
- A$\log _{e}\left(\frac{3}{2}\right)$
- B$\log _{e}\left(\frac{4}{3}\right)$
- C$\log _{e} 2$
- D$\log _{e} \sqrt{2}$
$\Rightarrow 4 \alpha\left(\left[\frac{\mathrm{e}^{\alpha \mathrm{x}}}{\alpha}\right]_{-1}^{0}+\left[\frac{\mathrm{e}^{-\alpha \mathrm{x}}}{-\alpha}\right]_{0}^{2}\right)=5$
$\Rightarrow 4 \mathrm{e}^{-2 \alpha}+4 \mathrm{e}^{-\alpha}-3=0$
Let $\mathrm{e}^{-\alpha}=\mathrm{t}, 4 \mathrm{t}^{2}+4 \mathrm{t}-3=0,$$ \mathrm{t}=\frac{1}{2}, \frac{-3}{2}$ (Rejected)
$\mathrm{e}^{-\alpha}=\frac{1}{2} ; \quad \alpha=\ln 2$
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