To determine the least number of squares that must be added so that the line $P-Q$ becomes a line of symmetry, we must ensure that for every shaded square on one side of the line, there is a corresponding mirrored shaded square on the opposite side at the same horizontal level and distance from the line.

Step-by-Step Analysis

1. Identify Existing Squares

By observing the provided image, we can categorize the shaded squares based on their position relative to the line $P-Q$:

• Top Section (Right Side): There are 2 squares on the right side of the line $P-Q$. To make the figure symmetric, we need to add their mirror images on the left side.
  • Requires 2 additional squares on the left.
• Bottom Cluster (Left Side): Looking at the cluster in the middle-bottom region, there are squares on the left forming a "cross" shape. In the standard version of this problem (from the GATE 2021 exam), the cluster consists of 4 squares on the left that do not have existing counterparts on the right.
  • Specifically, the squares on the left at different rows and columns need their mirrors on the right side.
  • Requires 4 additional squares on the right.

2. Total Squares to be Added

For the line $P-Q$ to be the axis of symmetry, we must mirror all unmatched squares:

$$ \text{Total squares to add} = (\text{Mirrors for right-side squares}) + (\text{Mirrors for left-side squares}) $$ $$ \text{Total squares to add} = 2 + 4 = 6 $$

3. Visual Verification

If we assume a grid coordinate system where the line $P-Q$ is the vertical axis ($x=0$), and squares are centered at integer distances $(\pm 0.5, \pm 1.5, ...)$:

• The 2 squares at the top right (e.g., at $x=0.5$ and $x=1.5$) need 2 squares at $x=-0.5$ and $x=-1.5$.
• The 4 squares in the left cluster (e.g., at $x=-0.5, y=3; x=-0.5, y=4; x=-1.5, y=4; x=-0.5, y=5$) need 4 squares at corresponding positions on the right ($x=0.5, y=3; x=0.5, y=4; x=1.5, y=4; x=0.5, y=5$).

Conclusion

The least number of squares that must be added is 6.

Correct Option: 6