Power in AC circuits

An inductor of 10 mH, capacitor of 0.1 μF and a resistor of 100 Ω are connected in series across an $a.c$ power supply 220 V, 70 Hz. The power factor of the given circuit is 0.5. The difference in the inductive reactance and capacitance reactance is $\sqrt3\,\alpha$ Ω. The value of $\alpha$ is ________.

A LCR series circuit driven with $E_{rms}=90$ V at frequency $f_d=30$ Hz has resistance $R=80$ Ω, an inductance with inductive reactance $X_L=20.0$ Ω and capacitance with capacitive reactance $X_C=80.0$ Ω. The power factor of the circuit is ________.

A series LCR circuit is connected to a $45\sin(\omega t)$ Volt source. The resonant angular frequency of the circuit is $10^5\ \mathrm{rad\,s^{-1}}$ and current amplitude at resonance is $I_0$. When the angular frequency of the source is $\omega = 8\times10^4\ \mathrm{rad\,s^{-1}}$, the current amplitude in the circuit is $0.05\,I_0$. If $L = 50$ mH, match each entry in List-I with an appropriate value from List-II and choose the correct option.

Given below are two statements :

Statement-I: The reactance of an ac circuit is zero. It is possible that the circuit contains a capacitor and an inductor.

Statement-II: In ac circuit, the average power delivered by the source never becomes zero.

In the light of the above statements, choose the correct answer from the options given below

If wattless current flows in the AC circuit, then the circuit is :

An AC source is connected to an inductance of 100 mH, a capacitance of 100 $\mu$F and a resistance of 120 $\Omega$ as shown in figure. The time in which the resistance having a thermal capacity 2 J/°C will get heated by 16°C is ______ s.

A series LCR circuit with $R=\dfrac{250}{11}$ $\Omega$ and $X_L=\dfrac{70}{11}$ $\Omega$ is connected across a 220 V, 50 Hz supply. The value of capacitance needed to maximize the average power of the circuit will be ______ $\mu$F. (Take : $\pi=\dfrac{22}7$)

In a series LR circuit $X_L=R$ and power factor of the circuit is $P_1$. When capacitor with capacitance C such that $X_L=X_C$ is put in series, the power factor becomes $P_2$. The ratio $\dfrac{P_1}{P_2}$ is :

To light, a 50 W, 100 V lamp is connected, in series with a capacitor of capacitance $\dfrac{50}{\pi\sqrt x}$ μF, with 200 V, 50 Hz AC source. The value of $x$ will be ______.

In an A.C circuit, V & I are given by V = 250 sin (100 t) volt and I = 10 sin $\left(100t+\dfrac\pi3\right)$ A respectively. The power dissipated in the circuit is:

The value of C is ____.