A series LCR circuit containing 5.0 H inductor, 80 μF capacitor and 40 $Ω$ resistor is connected to 230 V variable frequency ac sources. The angular frequencies of the source at which power transferred to the circuit is half the power at the resonant angular frequency are likely to be :

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46 rad/s and 54 rad/s

25 rad/s and 75 rad/s

50 rad/s and 25 rad/s

42 rad/s and 58 rad/s

answer is D.

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Detailed Solution

$\begin{array}{l} ω_{0} = \frac{1}{\sqrt{LC}} = \frac{1}{\sqrt{5 \times 80 \times 10^{- 6}}} \\ = \frac{1}{\sqrt{4 \times 10^{- 4}}} = \frac{10^{2}}{2} = 50 \end{array}$
Half power frequency band width is R/L
= 40/5 = 8
Half power frequencies
50 - 4 = 46 Hz and 50 + 4 = 54 Hz