$A$ 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 ............... $\text{J/s}$.

If a straight line $L$ is perpendicular to the line $4x - 2y = 1$ and forms a triangle of area $4 \text{ sq unit}$ with the coordinate axes,then the equation of the line $L$ is

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

The area (in sq unit) of the region bounded by the curves $2x = y^2 - 1$ and $x = 0$ is

$A$ meter bridge is set up as shown,to determine an unknown resistance '$X$' using a standard $10 \, \Omega$ resistor. The galvanometer shows a null point when the tapping-key is at the $52 \, cm$ mark. The end corrections are $1 \, cm$ and $2 \, cm$ respectively for the ends $A$ and $B$. The determined value of '$X$' is $.......... \Omega$.

The number of common tangents to the circles $x^2+y^2-4x-6y-12=0$ and $x^2+y^2+6x+18y+26=0$ is: