In $\triangle ABC$,$\frac{r_1-r}{a}+\frac{r_2-r}{b}+\frac{r_3-r}{c} =$

If $z = \frac{y}{x} \left[ \sin \frac{x}{y} + \cos \left( 1 + \frac{y}{x} \right) \right]$,then $x \frac{\partial z}{\partial x}$ is equal to

$A$ body of mass $M \ kg$ is at the top point of a smooth hemisphere of radius $5 \ m$. It is released to slide down the surface of the hemisphere. It leaves the surface when its velocity is $5 \ m/s$. At this instant,the angle made by the radius vector of the body with the vertical is: (Acceleration due to gravity $g = 10 \ m/s^2$) (in $^{\circ}$)

If $\log x : \log y : \log z = (y - z) : (z - x) : (x - y)$,then

$A$ steel rod $100 \, cm$ long is clamped at its middle. The fundamental frequency of longitudinal vibrations of the rod is $2.53 \, kHz$. What is the speed of sound in steel in $km/sec$?

In planetary motion,the areal velocity of the position vector of a planet depends on the angular velocity $\omega$ and the distance of the planet from the sun $r$. The correct relation for areal velocity is: