$A$ light bulb connected in series with a capacitor and an $a.c.$ source is glowing with certain brightness. On reducing the value of capacitance and frequency respectively, the brightness of the bulb

When an inductor $L$ and a resistor $R$ in series are connected across a $15 \, V, 50 \, Hz$ a.c. supply, a current of $0.3 \, A$ flows in the circuit. The current differs in phase from the applied voltage by $(\frac{\pi}{3})^c$. The value of $R$ is:
$(\sin \frac{\pi}{6} = \cos \frac{\pi}{3} = \frac{1}{2}, \sin \frac{\pi}{3} = \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2})$

$A$ coil has an inductance of $10 \, mH$. When it is connected to a $10 \, V \, dc$ source,the power dissipation is $20 \, W$. When it is connected to a $10 \, V \, ac$ source,the power dissipation is $10 \, W$. What is the frequency of the $ac$ source in $Hz$?

When a $DC$ voltage of $100 \, V$ is applied to an inductor, a $DC$ current of $5 \, A$ flows through it. When an $AC$ voltage of $200 \, V$ peak value is connected to the inductor, its inductive reactance is found to be $20\sqrt{3} \, \Omega$. The power dissipated in the circuit is . . . . . . $W$.

The variation of impedance $Z$ of a series $\text{LCR}$ circuit with the frequency $f$ of the source is shown in the figure. Which of the following statement$(s)$ is/are true?

Alternating current is flowing in an $LR$ circuit with inductance $L$ and resistance $R$. The frequency of the source is $f = \omega / 2\pi$. Which of the following statements is correct?