Locations P and Q are separated by a wide valley. The difference in levels of locations P and Q measured by a levelling instrument stationed near P is 3.0 m. The same instrument stationed near Q measures the difference in levels of locations P and Q as −1.0 m. Assume that the atmospheric refraction is same during the measurements. The true difference in levels (in m) of locations P and Q is

The correct option is 1.0.

To understand why this is correct, we can analyze the problem using the principles of reciprocal levelling. When measuring the difference in elevation between two points, P and Q, separated by a wide valley, two main sources of error affect the readings: the curvature of the Earth and atmospheric refraction. By placing the levelling instrument successively near P and then near Q, we can eliminate these errors.

Let:
- $h_P$ be the true elevation of location P.
- $h_Q$ be the true elevation of location Q.
- $H=h_P-h_Q$ be the true difference in levels between P and Q.
- $e$ be the combined error due to Earth's curvature and atmospheric refraction over the distance between P and Q. Since the atmospheric refraction is assumed to be the same during both measurements, this error remains constant.

When the instrument is stationed near P:
The backsight reading on P has negligible error because the instrument is very close to P. However, the foresight reading on Q, which is far away, includes the combined error $e$.
Thus, the measured difference in levels from P is:
$H_1=H+e=3.0\text{m}$

When the instrument is stationed near Q:
The reading on Q has negligible error, while the reading on P (far away) contains the same combined error $e$ but with the opposite effect on the difference since the direction of measurement is reversed.
Thus, the measured difference in levels from Q is:
$H_2=H-e=-1.0\text{m}$

To find the true difference in levels $H$, we take the average of the two measured differences:
$H=\frac{H_1+H_2}{2}$

Substituting the given values into the equation:
$H=\frac{3.0+(-1.0)}{2}$

Simplifying the expression yields:
$H=\frac{2.0}{2}=1.0\text{m}$

Therefore, the true difference in levels between locations P and Q is 1.0 m.