The lengths of the two pipes open at both ends are L and $(L + L_1)$. If they are sounded together, the beat frequency will be
($v$ = velocity of sound in air)

The number of waves counted by an observer on the sea-coast in one minute is 54. If the wavelength of the waves is $5\ \text{m}$ then the velocity of the waves in $\text{m/s}$ will be

When a dipole placed parallel to electric field is rotating through $\pi^c$, the work done is W. The work done in rotating the dipole through $\left(\dfrac{\pi}{3}\right)^c$ is
($\cos 0^\circ = 1$, $\cos 60^\circ = \dfrac{1}{2}$, $\cos 180^\circ = -1$)

The point charges $+q$, $-q$, $-q$, $+q$, $+Q$ and $-q$ are placed at the vertices of a regular hexagon ABCDEF as shown in the figure.

The electric field at the centre of the hexagon 'O' due to the five charges at A, B, C, D and F is thrice the electric field at centre 'O' due to charge $+Q$ at E alone. The value of Q is