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Is the statement in question true?

Work done by the source in one time period ($T$) is =$V_{\text{RMS}} × I_{\text{RMS}} ×\cos(x)×\frac{T}2$, where $x$ = angle between I phasor and V phasor.

In case of inductive or capacitive circuit as $x=+\frac{\pi}{2} \text{or} -\frac{\pi}{2}$, so work done by source is zero. How is current flowing then ?

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The instantaneous power is not zero. Sometimes is positive, sometimes is negative. But overall, for a period, is zero. Don't forget that the current goes back and forth.

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current without resistance does not carry any power. Only in resistors work is done. But your ideal circuit does not exist in reality, so it does not really matter that no power is spend-

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