Which physical quantity is conserved in both elastic and inelastic collisions?

$A$ slide with a frictionless curved surface,which becomes horizontal at its lower end,is fixed on the terrace of a building of height $3h$ from the ground,as shown in the figure. $A$ spherical ball of mass $m$ is released on the slide from rest at a height $h$ from the top of the terrace. The ball leaves the slide with a velocity $\vec{u}_0 = u_0 \hat{x}$ and falls on the ground at a distance $d$ from the building,making an angle $\theta$ with the horizontal. It bounces off with a velocity $\vec{v}$ and reaches a maximum height $h_1$. The acceleration due to gravity is $g$ and the coefficient of restitution of the ground is $e = 1 / \sqrt{3}$. Which of the following statement$(s)$ is(are) correct?
$(A)$ $\vec{u}_0 = \sqrt{2gh} \hat{x}$
$(B)$ $\vec{v} = \sqrt{2gh} \hat{x} + \sqrt{2gh} \hat{z}$
$(C)$ $\theta = 60^{\circ}$
$(D)$ $d / h_1 = 2\sqrt{3}$

$A$ ball of mass $2m$ is moving with velocity $v$ on a smooth surface and collides elastically head-on with another ball of mass $m$ which is at rest. If the ball of mass $m$ reaches up to the top of a frictionless elevated plane of height $h$,then the velocity $v$ of the heavy ball must be

Two identical blocks $A$ and $B$, each of mass $m$, resting on a smooth floor, are connected by a light spring of natural length $L$ and spring constant $k$. The spring is at its natural length. $A$ third identical block $C$ (mass $m$) moving with a speed $v$ along the line joining $A$ and $B$ collides with $A$. The maximum compression in the spring is proportional to

$A$ ball of mass $m$ is dropped from a height $s$ on a horizontal platform fixed at the top of a vertical spring. The platform is depressed by a distance $h$. The spring constant is ($g=$ acceleration due to gravity):

$A$ block of mass $m(=0.1 \ kg)$ is hanging over a frictionless light fixed pulley by an inextensible string of negligible mass. The other end of the string is pulled by a constant force $F$ in the vertically downward direction. The linear momentum of the block increases by $2 \ kg \ m/s$ in $1 \ s$ after the block starts from rest. Then, (given $g=10 \ m/s^2$):