The power dissipated in the circuit shown in the figure is $30 \, W$. The value of $R$ is ............. $\Omega$.

If $\int \frac{\sqrt{1-x^2}}{x^4} ~d x=A(x)\left(\sqrt{1-x^2}\right)^m+C$ for a suitable chosen integer $m$ and a function $A(x)$,where $C$ is a constant of integration,then $(A(x))^m$ equals

The energy of a photon is calculated by

$A$ musical instrument $P$ produces sound waves of frequency $n$ and amplitude $A_P$. Another musical instrument $Q$ produces sound waves of frequency $\frac{n}{4}$. The waves produced by $P$ and $Q$ have equal energies. If the amplitude of waves produced by $P$ is $A_P$,the amplitude of waves produced by $Q$ will be: (in $A_P$)

$\sum_{n=1}^{\infty} \frac{2n^2+n+1}{n!}$ is equal to

If $N$ denotes the set of all positive integers and if $f: N \rightarrow N$ is defined by $f(n) =$ the sum of positive divisors of $n$,then $f(2^k \cdot 3)$,where $k$ is a positive integer,is