A particle of mass $m$ moving with a velocity $\left(3\hat{\mathrm{i}}+2\hat{\mathrm{j}}\right) \mathrm{m} {\mathrm{s}}^{-1}$ collides with a stationary body of mass $M$ and finally moves with a velocity $\left(-2\hat{\mathrm{i}}+\hat{\mathrm{j}}\right) \mathrm{m} {\mathrm{s}}^{-1}$. If $\frac{m}{M}=\frac{1}{13}$, then

Two particles of masses ${\mathrm{m}}_1,\mathrm{ }{\mathrm{m}}_2$ move with initial velocities ${\mathrm{u}}_1$ and ${\mathrm{u}}_2$. On collision, one of the particles get excited to higher level, after absorbing energy $\mathrm{\epsilon .}$ If final velocities of particles be ${\mathrm{v}}_1$ and ${\mathrm{v}}_2$ then we must have:

A ball of mass $10 \mathrm{kg}$ moving with a velocity $10\sqrt{3} \mathrm{m} {\mathrm{s}}^{-1}$ along the $x$ -axis, hits another ball of mass $20 \mathrm{kg}$ which is at rest. After the collision, first ball comes to rest while the second ball disintegrates into two equal pieces. One piece starts moving along $y$ -axis with a speed of $10 \mathrm{m} {\mathrm{s}}^{-1}.$ The second piece starts moving at an angle of $30°$ with respect to the $x$ -axis. The velocity of the ball moving at $30°$ with $x$ -axis is $x \mathrm{m} {\mathrm{s}}^{-1}.$ The configuration of pieces after the collision is shown in the figure below. The value of $x$ to the nearest integer is