$A$ sphere is suspended by a thread of length $l$. What minimum horizontal velocity has to be imparted to the ball for it to reach the height of the suspension point?

$A$ fish looking up through the water sees the outside world,contained in a circular horizon. If the refractive index of water is $\frac{4}{3}$ and the fish is $12 \ cm$ below the water surface,the radius of this circle in $cm$ is

The value of $a$ for which the equations $x^3+ax+1=0$ and $x^4+ax^2+1=0$ have a common root is

If $u=\log \left(x^3+y^3+z^3-3 x y z\right)$,then $(x+y+z)(u_x+u_y+u_z)$ is equal to

Find the missing numbers in the following factorization tree:

$A$ particle undergoes uniform circular motion. About which point in the plane of the circle will the angular momentum of the particle remain conserved?