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(A) The time constant of the $RC$ circuit is $	au = RC = (1 	imes 10^3 \ \Omega) 	imes (10 	imes 10^{-9} \ 	ext{F}) = 10 	imes 10^{-6} \ 	ext{s

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(A) The time constant of the $RC$ circuit is $	au = RC = (1 	imes 10^3 \ \Omega) 	imes (10 	imes 10^{-9} \ 	ext{F}) = 10 	imes 10^{-6} \ 	ext{s} = 10 \ \mu	ext{s}$.During the first $5 \ \mu	ext{s}$ $(0 \le t \le 5 \ \mu	ext{s})$,the capacitor charges from $0 \ 	ext{V}$ towards $5 \ 	ext{V}$ according to $V_0(t) = 5(1 - e^{-t/	au})$.At $t = 5 \ \mu	ext{s}$,$V_0(5 \ \mu	ext{s}) = 5(1 - e^{-5/10}) = 5(1 - e^{-0.5}) \approx 5(1 - 0.6065) = 5(0.3935) \approx 1.9675 \ 	ext{V} \approx 2 \ 	ext{V}$.During the next $5 \ \mu	ext{s}$ $(5 \ \mu	ext{s} \le t \le 10 \ \mu	ext{s})$,the input is $0 \ 	ext{V}$,so the capacitor discharges from $2 \ 	ext{V}$ towards $0 \ 	ext{V}$ according to $V_0(t) = 2e^{-(t-5)/	au}$.At $t = 10 \ \mu	ext{s}$,$V_0(10 \ \mu	ext{s}) = 2e^{-(10-5)/10} = 2e^{-0.5} \approx 2(0.6065) \approx 1.213 \ 	ext{V}$.During the next $5 \ \mu	ext{s}$ $(10 \ \mu	ext{s} \le t \le 15 \ \mu	ext{s})$,the capacitor charges from $1.213 \ 	ext{V}$ towards $5 \ 	ext{V}$ according to $V_0(t) = 5 - (5 - 1.213)e^{-(t-10)/	au} = 5 - 3.787e^{-(t-10)/	au}$.At $t = 15 \ \mu	ext{s}$,$V_0(15 \ \mu	ext{s}) = 5 - 3.787e^{-0.5} \approx 5 - 3.787(0.6065) \approx 5 - 2.297 = 2.703 \ 	ext{V} \approx 2.7 \ 	ext{V}$.Comparing this behavior with the given options,Option $A$ correctly shows the charging and discharging cycles with the calculated values.

For the given input voltage waveform $V_{\text{in}}(t)$,the output voltage waveform $V_0(t)$ across the capacitor is correctly depicted by:

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