If $\tanh ^{-1} x = a \log \left(\frac{1+x}{1-x}\right)$,$|x| < 1$,then $a$ is equal to

The wavelengths of two notes in air are $\frac{36}{195} ~m$ and $\frac{36}{193} ~m$. Each note produces $10$ beats per second separately with a third note of fixed frequency. The velocity of sound in air in $m/s$ is

The area of the triangle formed by the lines ${y^2} - 9xy + 18{x^2} = 0$ and $y = 9$ is

In $\triangle ABC$,with usual notation,observe the two statements given below:
$(I)$ $r r_1 r_2 r_3 = \Delta^2$
$(II)$ $r_1 r_2 + r_2 r_3 + r_3 r_1 = s^2$
Which of the following is correct?

Crotonyl alcohol is an example of

Two equally charged metal spheres $A$ and $B$ repel each other with a force of $4 \times 10^{-5} ~N$. Another identical uncharged sphere $C$ is touched to $A$ and then placed at the midpoint of the line joining the spheres $A$ and $B$. The net electric force on the sphere $C$ is