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Power factor of an LR circuit before and after adding a capacitor: ratio

The same question, asked 3 times: 2021 Paper 21*, 26 Jul 2022, 30 Jan 2023. Every wording below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    2021 Paper 21

    27
    An ac circuit has an inductor and a resistor of resistance RR in series, such that XL=3RX_{L}= 3R. Now, a capacitor is added in series such that XC=2RX_{C}= 2R. The ratio of new power factor with the old power factor of the circuit is 5:x\sqrt{5}:x. The value of xx is

  2. Question 2

    26 Jul 2022

    7
    In a series LRLR circuit XL=RX_{L}= R and power factor of the circuit is P1P_{1}. When capacitor with capacitance CC such that XL=XCX_{L}=X_{C} is put in series, the power factor becomes P2P_{2}. The ratio P1P2\frac{P_{1}}{P_{2}} is

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  3. Question 3

    30 Jan 2023

    10
    In a series LR circuit with XL=RX_{L}= R, power factor is P1P_{1}. If a capacitor of capacitance CC with XC=XLX_{C}=X_{L} is added to the circuit the power factor becomes P2P_{2}. The ratio of P1P_{1} to P2P_{2} will be:

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