1. Original resonant frequency for a series RLC circuit is $f_1 = \frac{1}{2\pi\sqrt{LC}}$

2. If both inductance and capacitance are doubled, then $L' = 2L, \qquad C' = 2C$

   So the new resonant frequency is $f_2 = \frac{1}{2\pi\sqrt{L'C'}} = \frac{1}{2\pi\sqrt{(2L)(2C)}}$ $f_2 = \frac{1}{2\pi\sqrt{4LC}} = \frac{1}{2\pi \cdot 2\sqrt{LC}}$ $f_2 = \frac{1}{2} \cdot \frac{1}{2\pi\sqrt{LC}} = \frac{f_1}{2}$

3. Quality factor for a series RLC circuit is $Q = \frac{1}{R}\sqrt{\frac{L}{C}}$

   Original quality factor: $Q_1 = \frac{1}{R}\sqrt{\frac{L}{C}}$

   New quality factor: $Q_2 = \frac{1}{R}\sqrt{\frac{L'}{C'}} = \frac{1}{R}\sqrt{\frac{2L}{2C}}$ $Q_2 = \frac{1}{R}\sqrt{\frac{L}{C}} = Q_1$

4. Therefore, $f_2 = \frac{f_1}{2}, \qquad Q_2 = Q_1$

5. Option check

   • A: $f_2 = \dfrac{f_1}{2}$ and $Q_2 = Q_1$ ✔️
   • B: Incorrect
   • C: Incorrect
   • D: Incorrect

Hence, the correct option is A.