An LCR series circuit containing resistance $R_1$, inductance $L_1$ and capacitance $C_1$ gives resonance at the same frequency $f$ as a second similar combination $R_2, L_2$ and $C_2$. If the two circuits are connected in series, then the frequency of the combined circuit is.

$f$

The correct answer is Option (3) → $f$

Equivalent capacitance in series = $\frac{C_1 C_2}{C_1+C_2}$

Equivalent inductance in series = $L_1+L_2$

Frequency of combined circuit $\omega=\sqrt{\frac{1}{L C}}$

$=\sqrt{\frac{C_1+C_2}{C_1 C_2\left(L_1+L_2\right)}}$

$=\sqrt{\frac{C_1+C_2}{L_1 C_1 C_2+L_2 C_1 C_2}}$

As frequency of individual circuit is same $L_1 C_1=L_2 C_2$.

$\omega=\sqrt{\frac{C_1+C_2}{L_1 C_1 C_2+L_1 C_1 C_1}}$

$=\sqrt{\frac{C_1+C_2}{L_1 C_1\left(C_1+C_2\right)}}$

$=\sqrt{\frac{1}{L_1 C_1}}$

$\omega=\omega$ of individual circuit.

So, frequency is also same 

(3) is correct ans.