$A$ sinusoidal carrier voltage of frequency $1.5\,MHz$ and amplitude $50\,V$ is amplitude modulated by a sinusoidal voltage of frequency $10\,kHz$ producing $50\%$ modulation. The lower and upper side-band frequencies in $kHz$ are:

In a circuit,$L$,$C$,and $R$ are connected in series with an alternating voltage source of frequency $f$. The current leads the voltage by $45^{\circ}$. The value of $C$ is

The half-life of a radioactive substance is $20 \text{ minutes}$. The approximate time interval $(t_2 - t_1)$ between the time $t_2$ when $3/4$ of it has decayed and time $t_1$ when $1/4$ of it has decayed is:

If a set $A$ has $n$ elements,then the total number of subsets of $A$ is:

Two circles whose radii are equal to $4$ and $8$ intersect at right angles. The length of their common chord is

When metal $X$ is treated with sodium hydroxide,a white precipitate $(A)$ is obtained,which is soluble in excess of $NaOH$ to give soluble complex $(B)$. Compound $(A)$ is soluble in dilute $HCl$ to form compound $(C)$. The compound $(A)$ when heated strongly gives $(D)$,which is used to extract metal. Identify $(X), (A), (B), (C)$ and $(D)$. Write suitable equations to support their identities.