1. Given data

• Frequency: $f = 750\,\text{Hz}$
• RMS voltage: $V = 20\,\text{V}$
• Resistance: $R = 100\,\Omega$
• Inductance: $L = 0.1803\,\text{H}$
• Capacitance: $C = 10\,\mu\text{F} = 10\times 10^{-6}\,\text{F}$
• Heat capacity of resistance: $2\,\text{J}/^\circ\text{C}$
• Required temperature rise: $10^\circ\text{C}$

So, heat required is $Q = (2)(10) = 20\,\text{J}.$

2. Find angular frequency

$\omega = 2\pi f = 2\pi(750) = 1500\pi\,\text{rad/s}.$

Numerically, $\omega \approx 4712.39\,\text{rad/s}.$

3. Calculate reactances

Inductive reactance: $X_L = \omega L = (4712.39)(0.1803) \approx 849.34\,\Omega.$

Capacitive reactance: $X_C = \frac{1}{\omega C} = \frac{1}{(4712.39)(10\times10^{-6})} \approx 21.22\,\Omega.$

Net reactance: $X = X_L - X_C = 849.34 - 21.22 = 828.12\,\Omega.$

4. Find impedance of series RLC circuit

$Z = \sqrt{R^2 + (X_L-X_C)^2}$ $Z = \sqrt{100^2 + 828.12^2}$ $Z \approx \sqrt{10000 + 685786.73}$ $Z \approx \sqrt{695786.73} \approx 834.14\,\Omega.$

5. Find current in the circuit

$I = \frac{V}{Z} = \frac{20}{834.14} \approx 0.02398\,\text{A}.$

6. Power dissipated in the resistor

Only the resistor dissipates real power: $P = I^2R$ $P = (0.02398)^2(100) \approx 0.0575\,\text{W}.$

7. Find time required to produce 20 J heat

Since $Q = Pt,$ we get $t = \frac{Q}{P} = \frac{20}{0.0575} \approx 347.8\,\text{s}.$

Thus the required time is close to $\boxed{348\,\text{s}}.$

8. Option check

• A: $348\,\text{s}$ ✅
• B: $418\,\text{s}$ ❌
• C: $245\,\text{s}$ ❌
• D: $365\,\text{s}$ ❌

Therefore, the correct option is A.