In the given circuit,find the equivalent resistance between points $a$ and the combination of $b$ and $c$ (where $b$ and $c$ are connected together).

Two simple pendulums whose lengths are $1\,m$ and $121\,cm$ are suspended side by side. Their bobs are pulled together and then released. After how many minimum oscillations of the longer pendulum will the two be in phase again?

Pure ammonia is placed in a vessel at a temperature where its dissociation constant $(\alpha)$ is appreciable. At equilibrium,

Shown in the figure is a circular loop of radius $r$ and resistance $R$. $A$ variable magnetic field of induction $B = B_0e^{-t}$ is established inside the coil. If the key $(K)$ is closed,the electrical power developed right after closing the switch is equal to

In the circuit shown,the reading of the Ammeter is doubled after the switch is closed. Each resistor has a resistance $= 1\,\Omega$ and the ideal cell has an $e.m.f. = 10\,V$. Then,the Ammeter has a coil resistance equal to .............. $\Omega$.

$\frac{\sin 1^{\circ}+\sin 2^{\circ}+\ldots+\sin 89^{\circ}}{2(\cos 1^{\circ}+\cos 2^{\circ}+\ldots+\cos 44^{\circ})+1} = $