**Solution:**

To solve this problem, we need to consider the motion of the charged particle P in the presence of a magnetic field B. We can use the Lorentz force equation to determine the trajectory of the particle.

The Lorentz force equation is given by:

**F = q(v x B)**

where F is the force experienced by the charged particle, q is the charge of the particle, v is the velocity of the particle, and B is the magnetic field.

**1. Initial Motion of Particle P**

Before we consider the minimum value of B, let's analyze the initial motion of the particle P. The particle is leaving the origin with speed v₀ at some inclination with the X-axis. Since there is a uniform magnetic field along the X-axis, the force experienced by the particle will be perpendicular to both the velocity and the magnetic field.

**2. Force Experienced by Particle P**

The force experienced by the particle P can be calculated using the Lorentz force equation. Since the force is perpendicular to the velocity, it will produce a centripetal motion, causing the particle to move in a circular path in the X-Y plane.

**3. Minimum Value of B**

To determine the minimum value of B for the particle P to strike the fixed target T on the X-axis, we need to analyze the trajectory of the particle. Since the target is fixed on the X-axis, the particle should travel in a straight line from the origin to the target.

**4. Factors Affecting Particle's Trajectory**

The trajectory of the particle is affected by the velocity and the magnetic field. To strike the target T, the particle should move in a straight line, which means the centripetal force should be zero. This can be achieved if the magnetic force is balanced by the centrifugal force.

**5. Magnetic Force and Centrifugal Force**

The magnetic force experienced by the particle is given by:

**F = q(v x B)**

The centrifugal force experienced by the particle is given by:

**F = m(v²/R)**

where m is the mass of the particle and R is the radius of the circular path.

**6. Balancing the Forces**

To balance the forces, we can equate the magnetic force and the centrifugal force:

**q(v x B) = m(v²/R)**

Since the particle is moving in a straight line, the radius of the circular path is infinite, and the centrifugal force becomes zero. Therefore, the magnetic force should also become zero.

**7. Minimum Value of B**

For the magnetic force to become zero, the cross product of the velocity and the magnetic field should be zero. This can happen if the angle between the velocity and the magnetic field is zero or 180 degrees.

**8. Angle Between Velocity and Magnetic Field**

The angle between the velocity and the magnetic field can be calculated using the dot product:

**v · B = |v| |B| cos(theta)**

where theta is the angle between the velocity and the magnetic field.

If cos(theta) = 1, then the angle is zero, and if cos(theta) = -1, then the angle is 180 degrees.

**9. Conclusion**

To summarize, the charged particle P will strike the fixed target T on the X-axis for the minimum value of B = B₀ if the velocity v and the magnetic field B are such that the angle between them is