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

$A$ ray of light of intensity $I$ is incident on a parallel glass slab at point $A$ as shown in the diagram. It undergoes partial reflection and refraction. At each reflection, $25\%$ of incident energy is reflected. The rays $AB$ and $A'B'$ undergo interference. The ratio of $I_{max}$ and $I_{min}$ is (in $: 1$)

Which of the following methods is generally used for the extraction of alkali metals?

$(a)$ Show that for every natural number $n$ relatively prime to $10$,there exists a natural number $m$ consisting only of digits $1$ such that $n$ divides $m$.
$(b)$ Hence or otherwise,show that every positive rational number can be expressed in the form $\frac{a}{10^b(10^c-1)}$ for some natural numbers $a, b, c$.

The major product of the following reaction is:

The numbers $a_n = 6^n - 5n$ for $n = 1, 2, 3, \ldots$ when divided by $25$ leave the remainder: