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(A, C) The particle enters region-$c$ if the radius of its circular path $R$ is greater than the width $L$ of region-$b$.Since $R = \frac{mv}{qB}$, th

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Time spent in region-$b$ is independent of velocity $v$

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(A, C) The particle enters region-$c$ if the radius of its circular path $R$ is greater than the width $L$ of region-$b$.Since $R = \frac{mv}{qB}$, the condition for entering region-$c$ is $\frac{mv}{qB} > L$, which implies $v > \frac{qBL}{m}$. Thus, option $A$ is correct.Since the magnetic field $B$ is uniform and the velocity vector $\vec{v}$ is perpendicular to the magnetic field $\vec{B}$, the magnetic force provides the necessary centripetal force, making the path of the particle circular in region-$b$. Thus, option $C$ is correct.The time spent in region-$b$ is given by $t = \frac{	heta}{\omega}$, where $	heta$ is the angle subtended by the arc at the center and $\omega = \frac{qB}{m}$ is the angular velocity. Since the angle $	heta$ depends on the radius $R$ (and thus on velocity $v$), the time spent in region-$b$ depends on the velocity $v$. Therefore, option $D$ is incorrect.

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