If the extremities of a diagonal of a square are $(1, -2, 3)$ and $(2, -3, 5)$,then the length of its side is

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:

From a uniform wire,two circular loops are made: $(i)$ $P$ of radius $r$ and $(ii)$ $Q$ of radius $nr$. If the moment of inertia of $Q$ about an axis passing through its centre and perpendicular to its plane is $8$ times that of $P$ about a similar axis,the value of $n$ is (diameter of the wire is very much smaller than $r$ or $nr$).

$A$ bucket tied at the end of a $1.6\, m$ long string is whirled in a vertical circle with constant speed. What should be the minimum speed so that the water from the bucket does not spill,when the bucket is at the highest position? (Take $g = 10\, m/sec^2$)

$x = \frac{1-\sqrt{y}}{1+\sqrt{y}} \Rightarrow \frac{dy}{dx}$ is equal to

In a $\triangle ABC$,if $\frac{a}{b^2-c^2} + \frac{c}{b^2-a^2} = 0$,then $B$ is equal to