The correct answer is: A. error due to horizontal axis not being perpendicular to the vertical axis.
The other options are errors that can be eliminated by taking both face observations.
- Index error is the error due to imperfect adjustment of the vertical circle vernier. It can be eliminated by taking the mean of the two face observations.
- Error due to non-parallelism of the axis of telescope level and line of collimation can be eliminated by taking the mean of the two face observations.
The error due to horizontal axis not being perpendicular to the vertical axis cannot be eliminated by taking both face observations. This is because the error is caused by the misalignment of the horizontal axis and the vertical axis of the theodolite. When the theodolite is set up, the horizontal axis should be perpendicular to the vertical axis. However, due to manufacturing errors or improper setup, the horizontal axis may not be perfectly perpendicular to the vertical axis. This will cause an error in the measurement of the angle between two lines. The error will be the same for both face observations, so it cannot be eliminated by taking the mean of the two face observations.
To eliminate this error, the theodolite should be checked for proper alignment. If the horizontal axis is not perpendicular to the vertical axis, the theodolite should be adjusted or repaired.