In damped oscillations, damping force is directly proportional to speed of oscillator. If amplitude becomes half of its maximum value in $1 \,s$, then after $2 \,s$ amplitude will be $\left(A_0-\right.$ initial amplitude)
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(a)
$A=A_0 e^{-b t}$
Amplitude becomes half hence
$\frac{A_0}{2} A_0 e^{-b t}[t=1]$
$\therefore e^{-b}=\frac{1}{2}$
$\therefore \text { In two seconds }$
$A=A_0\left(\frac{1}{2}\right)^2$
$A=\frac{A_0}{4}$
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