A particle of charge q and mass m moving with a velocity v along x−axis enters the region x>0 with uniform magnetic field B along the k direction. What is the penetration of the particle in this region in the x−direction up to a distance d?

2mv/qB

The particle move in a circle, the plane of the circle is perpendicular to the direction of magnetic field.
Centripetal force is provided  by the  Lorentz force.

so, qvB = mv²/r

and r= mv/qB

Penetration is the diameter, i.e. 2r.