An athlete completes one round of a circular track of radius $R$ in $40 \ s$. What will be his displacement at the end of $2 \ min \ 20 \ s$?

The chemicals and the reaction conditions required for the preparation of ethane are:

The equation of the pair of straight lines parallel to the $X$-axis and touching the circle $x^2+y^2-6x-4y-12=0$ is:

The spring balance $A$ reads $2 \ kg$ with a block $m$ suspended from it. $A$ balance $B$ reads $5 \ kg$ when a beaker with liquid is placed on the pan of the balance. The two balances are now arranged such that the hanging mass is inside the liquid in the beaker as shown in the figure. In this situation:

The magnetic field $(B)$ inside a long solenoid having $n$ turns per unit length and carrying current $I$ when an iron core is kept in it is ($\mu_0 = $ permeability of vacuum,$\chi = $ magnetic susceptibility).

An $AC$ source of angular frequency $\omega$ is fed across a resistor $R$ and a capacitor $C$ in series. The current registered is $I$. If now the frequency of the source is changed to $\omega/3$ (but maintaining the same voltage),the current in the circuit is found to be halved. Calculate the ratio of reactance to resistance at the original frequency $\omega$.