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| column $I$ | column $II$ |
| $(A)$ $U _1( x )=\frac{ U _0}{2}\left[1-\left(\frac{ x }{ a }\right)^2\right]^2$ | $(P)$ The force acting on the particle is zero at $x = a$. |
| $(B)$ $U _2( x )=\frac{ U _0}{2}\left(\frac{ x }{ a }\right)^2$ | $(Q)$ The force acting on the particle is zero at $x=0$. |
| $(C)$ $U _3( x )=\frac{ U _0}{2}\left(\frac{ x }{ a }\right)^2 \exp \left[-\left(\frac{ x }{ a }\right)^2\right]$ | $(R)$ The force acting on the particle is zero at $x =- a$. |
| $(D)$ $U _4( x )=\frac{ U _0}{2}\left[\frac{ x }{ a }-\frac{1}{3}\left(\frac{ x }{ a }\right)^3\right]$ | $(S)$ The particle experiences an attractive force towards $x =0$ in the region $| x |< a$. |
| $(T)$ The particle with total energy $\frac{ U _0}{4}$ can oscillate about the point $x=-a$. |
$(A)$ $\beta=0$ when $a= g / \sqrt{2}$
$(B)$ $\beta>0$ when $a= g / \sqrt{2}$
$(C)$ $\beta=\frac{\sqrt{2}-1}{\sqrt{2}}$ when $a= g / 2$
$(D)$ $\beta=\frac{1}{\sqrt{2}}$ when $a= g / 2$