Let $x \in C ^ { 2 } \left( \left[ 0 , + \infty \left[ ; \mathbb { R } ^ { n } \right) \right. \right.$ be a solution of the differential equation (1). For each real $t \geq 0$ we set, $T \left( x ^ { \prime } \right) ( t ) = \frac { 1 } { 2 } \left\langle A x ^ { \prime } ( t ) ; x ^ { \prime } ( t ) \right\rangle$ and $U ( x ) ( t ) = \frac { 1 } { 2 } \langle K x ( t ) ; x ( t ) \rangle$. Show then that the quantity $T \left( x ^ { \prime } \right) ( t ) + U ( x ) ( t )$ does not depend on $t \in [ 0 , + \infty [$.

The potential energy of a 1 kg particle free to move along the $x$-axis is given by $$V(x) = \left(\frac{x^4}{4} - \frac{x^2}{2}\right) \text{J}$$ The total mechanical energy of the particle is 2 J. Then, the maximum speed (in m/s) is
(1) 2
(2) $3/\sqrt{2}$
(3) $\sqrt{2}$
(4) $1/\sqrt{2}$

A particle is moving in a circle of radius $r$ under the action of a force $F = \alpha r ^ { 2 }$ which is directed towards centre of the circle. Total mechanical energy (kinetic energy + potential energy) of the particle is (take potential energy $= 0$ for $r = 0$ ):
(1) $\frac { 5 } { 6 } \alpha r ^ { 3 }$
(2) $\alpha r ^ { 3 }$
(3) $\frac { 1 } { 2 } \alpha r ^ { 3 }$
(4) $\frac { 4 } { 3 } \alpha r ^ { 3 }$

As shown in the figure, a particle of mass 10 kg is placed at a point $A$. When the particle is slightly displaced to its right, it starts moving and reaches the point $B$. The speed of the particle at $B$ is $x \mathrm {~m} \mathrm {~s} ^ { - 1 }$. (Take $g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$) The value of $x$ to the nearest integer is [Figure]

Given below is the plot of a potential energy function $\mathrm { U } ( \mathrm { x } )$ for a system, in which a particle is in one dimensional motion, while a conservative force $\mathrm { F } ( \mathrm { x } )$ acts on it. Suppose that $\mathrm { E } _ { \text {mech} } = 8 \mathrm {~J}$, the incorrect statement for this system is:
[where K.E. = kinetic energy]
(1) at $\mathrm { x } > \mathrm { x } _ { 4 }$, K. E. is constant throughout the region.
(2) at $\mathrm { x } < \mathrm { x } _ { 1 }$, K. E. is smallest and the particle is moving at the slowest speed.
(3) at $\mathbf { x } = \mathbf { x } _ { 2 }$, K. E. is greatest and the particle is moving at the fastest speed.
(4) at $x = x _ { 3 }$, K.E. $= 4 \mathrm {~J}$