Three objects, $A :$ (a solid sphere), $B :$ (a thin circular disk) and $C :$ (a circular ring), each have the same mass $M$ and radius $R.$ They all spin with the same angular speed $\omega$ about their own symmetry axes. The amounts of work $(W)$ required to bring them to rest, would satisfy the relation

A rolling wheel of $12 \,kg$ is on an inclined plane at position $P$ and connected to a mass of $3 \,kg$ through a string of fixed length and pulley as shown in figure. Consider $PR$ as friction free surface. The velocity of centre of mass of the wheel when it reaches at the bottom $Q$ of the inclined plane $P Q$ will be $\frac{1}{2} \sqrt{ xgh } \,m / s$. The value of $x$ is.............

A solid sphere rolls down without slipping on an inclined plane, then percentage of rotational kinetic energy of total energy will be ........ $\%.$

Two identical circular loops are moving with same kinetic energy one rolls $\&$ other slides. The ratio of their speed is 

The angular velocity of a body is $\mathop \omega \limits^ \to   = 2\hat i + 3\hat j + 4\hat k$ and a torque $\mathop \tau \limits^ \to   = \hat i + 2\hat j + 3\hat k$ acts on it. The rotational power will be .......... $W$

This question has Statement $1$ and Statement $2$. Of the four choices given after the Statements, choose the one that best describes the two Statements.
Statement $1$ : When moment of inertia $I$ of a body rotating about an axis with angular speed $\omega $ increases, its angular momentum $L$ is unchanged but the kinetic energy $K$ increases if there is no torque applied on it.
Statement $2$ : $L = I\omega $, kinetic energy of rotation $ = \frac{1}{2}\,I\omega ^2$