Step-by-step Solution:

1. Analyze the experimental setup and observation. The experiment uses the u-v method to find the focal length of a concave mirror. An object pin A is placed at a distance x from the pole P. A real, inverted image is observed. The method of parallax is used to compare the positions of the object and the image.

2. Understand the parallax method. The rule of parallax states that when an observer moves their head or eye sideways, the object that is closer to the eye appears to move in the direction opposite to the eye's movement, relative to an object that is farther away. The farther object appears to move in the same direction as the eye's movement.

3. Apply the parallax rule to the given information.

   • The student shifts their eye towards the left.
   • The image appears to move to the right relative to the object pin.
   • The direction of the image's apparent motion (right) is opposite to the direction of the eye's motion (left).
   • According to the parallax rule, this means the image is closer to the observer's eye than the object pin is.

4. Relate the observation to the distances from the mirror. In this setup, the student's eye is on the principal axis, looking towards the mirror, and is farther away from the mirror than both the object and its real image. If the image is closer to the eye, it must be farther from the mirror's pole.

   • Let x be the distance of the object from the pole (|u| = x).
   • Let v be the distance of the image from the pole (|v|).
   • The conclusion from step 3 is that the image is farther from the mirror than the object. Therefore, v > x.

5. Use the properties of a concave mirror and the mirror formula.

   • The problem states the image is inverted. For a concave mirror, an inverted image is always a real image.
   • A real image is formed only when the object is placed beyond the focal point (F). Thus, the object distance x must be greater than the focal length f. $x > f$
   • The mirror formula for a concave mirror forming a real image, using magnitudes for distances, is: $\frac{1}{v} + \frac{1}{x} = \frac{1}{f}$

6. Combine the parallax result with the mirror formula.

   • From step 4, we have the inequality v > x.
   • Since v and x are positive distances, taking the reciprocal reverses the inequality sign: 1/v < 1/x.
   • Now, let's use the mirror formula 1/v = 1/f - 1/x and substitute it into the inequality 1/v < 1/x: $\frac{1}{f} - \frac{1}{x} < \frac{1}{x}$
   • Rearranging the inequality to solve for x: $\frac{1}{f} < \frac{2}{x}$ $x < 2f$

7. Determine the final range for x.

   • From step 5, we know x > f (for a real image to form).
   • From step 6, we know x < 2f (from the parallax observation).
   • Combining these two conditions gives the final range for the object distance x: $f < x < 2f$

8. Conclusion. The object pin is placed between the focus (F) and the center of curvature (C) of the concave mirror. This corresponds to option B.