The number of natural numbers less than $1000$,in which no two digits are repeated,is:

$A$ particle moves with a velocity $\vec{v} = 5\hat{i} - 3\hat{j} + 6\hat{k} \text{ m/s}$ under the influence of a constant force $\vec{F} = 10\hat{i} + 10\hat{j} + 20\hat{k} \text{ N}$. The instantaneous power will be ............... $\text{J/s}$.

The locus of a point such that the sum of its distances from the points $(0, 2)$ and $(0, -2)$ is $6$,is

Considering only the Principal values of inverse functions,the set $A = \{x \geq 0 \mid \tan^{-1}(2x) + \tan^{-1}(3x) = \frac{\pi}{4}\}$

Three point charges $1 \ C$,$-2 \ C$,and $-2 \ C$ are placed at the vertices of an equilateral triangle of side $1 \ m$. The work done by an external force to increase the separation of the charges to $2 \ m$ in joules is $(\varepsilon_0 = \text{permittivity of air})$

$A$ radioactive material decays by simultaneous emission of two particles with respective half-lives $1620 \ years$ and $810 \ years$. The time (in years) after which one-fourth of the material remains is