A transistor-oscillator using a resonant circuit with an inductor $$L$$ (of negligible resistance) and a capacitor $$C$$ in series produce oscillations of frequency $$f.$$ If $$L$$ is doubled and $$C$$ is changed to $$4C,$$  the frequency will be

Solution :
We know that frequency of electrical oscillation in $$L.C.$$  circuit is
$$\eqalign{ & f = \frac{1}{{2\pi }}\sqrt {\frac{1}{{LC}}} \cr & {\text{Now,}}\,L = 2L\,\,\& \,\,C = 4C \cr & f' = \frac{1}{{2\pi }}\sqrt {\frac{1}{{2L \cdot 4C}}} = \frac{1}{{2\pi }}\sqrt {\frac{1}{{LC}}} \times \frac{1}{{2\sqrt 2 }} \Rightarrow f' \cr & = \frac{1}{{2\sqrt 2 }} \times f \cr} $$