In an ac circuit, the instantaneous e.m.f. and current are given by
\(\begin{aligned} & e=100 \sin 30 t \\ & i=20 \sin \left(30 t-\frac{\pi}{4}\right) \end{aligned}\)
In one cycle of ac, the average power consumed by the circuit and the wattless current are, respectively:
1. \(50, 10\)
2. \(\frac{1000}{\sqrt{2}},10\)
3. \(\frac{50}{\sqrt{2}},0\)
4. \(50,0\)

A \(750~\text{Hz},\) \(20~\text V\) (RMS) source is connected to a resistance of \(100~\Omega,\) an inductance of \(0.1803~\text H\) and a capacitance of \(10~{\mu \text{F}},\) all in series. The time in which the resistance (heat capacity \(2~\text J/^\circ \text C\)) will get heated by \(10^\circ \text {C}\) (assuming no loss of heat to the surroundings) is close to:
1. \(365~\text{s}\)
2. \(418~\text{s}\)
3. \(245~\text{s}\)
4. \(348~\text{s}\)

In a series \(LR\) circuit, a power of \(400\) W is dissipated from a source of \(250\) V and \(50\) Hz. The power factor of the circuit is \(0.8 \cdot\) To bring the power factor to unity, a capacitor of value \(\left(\dfrac{n}{3 \pi}\right) ~\mu \text{F}\)​ is added in series with the \(L\) and \(R\) components. The value of \(n\) is:

In a series \(LCR\) resonant circuit, the quality factor \((Q)\) at resonance is measured as \(100\). The inductance \((L)\) is increased two-fold \((L'= 2L)\), and the resistance \((R)\) is decreased two-fold \(\left(R' = \frac{R}{2}\right)\), while the capacitance \((C)\) remains unchanged. Assuming the new quality factor is calculated at the new resonance frequency of the modified circuit, then the new quality factor will be:
1. \(173.25\) 
2. \(282.84\)
3. \(453.97\)
4. \(621.24\)