The length of the subtangent at any point of the curve $x^m y^n = a^{m+n}$ is proportional to -

The number of $5$-digit numbers that are not divisible by $5$ and consist of different odd digits is:

$A$ $1 \, kg$ diatomic gas is at a pressure of $8 \times 10^4 \, N/m^2$. The density of the gas is $4 \, kg/m^3$. What will be the energy associated with the thermal motion of the gas?

Four resistors $A, B, C$ and $D$ form a Wheatstone bridge. The bridge is balanced when $C = 100 \ \Omega$. If $A$ and $B$ are interchanged,the bridge balances for $C = 121 \ \Omega$. The value of $D$ is (in $\Omega$)

If $x_n = \cos \frac{\pi}{2^n} + i \sin \frac{\pi}{2^n}$,then $\prod_{n=1}^{\infty} x_n$ is equal to

If $\theta$ lies in the first quadrant and $5 \tan \theta = 4$,then $\frac{5 \sin \theta - 3 \cos \theta}{\sin \theta + 2 \cos \theta}$ is equal to