GATE 2022 EC GA Q.5: solution and answer

Official answer: CSource: Official answer key, GATE 2022 EC (IIT Kharagpur)

1. The guard can see a point $X$ of the gallery only if the straight segment from the guard to $X$ stays inside the shaded region and never crosses a wall. We need every posting from which this holds for all points of the shaded region.

2. Name the corners of the plan: $A$ is the top-left tip, $C$ is the top-right tip, $B$ is the bottom tip, and $N$ is the inward notch. The gallery is the quadrilateral $ANCB$, and $N$ is its only inward-pointing corner. The segment $NB$ runs from the notch straight down the centre line to the bottom tip. It lies inside the gallery and cuts it into two triangles: $ANB$ (the left arm and the left half of the stem) and $NCB$ (the right arm and the right half of the stem).

3. A triangle is convex: the segment joining any two of its points stays inside it. Q and S both lie on the centre line $NB$, so each belongs to both triangles. Every point $X$ of the gallery lies in $ANB$ or in $NCB$. The segment from Q (or from S) to $X$ therefore stays inside that triangle, and so inside the gallery. With a $360^{\circ }$ view, Q sees the entire inner space, and so does S.

4. P and R are in the upper parts of the two arms at about the same height, both above the notch $N$. The straight line from P to R passes above $N$, through the unshaded V-shaped gap between the arms, so it crosses the inner wall $AN$ and then the inner wall $NC$. A guard at P therefore cannot see the point R, which is part of the shaded region. By the same line, a guard at R cannot see the point P. Neither P nor R is a valid posting.

5. The set of ALL valid postings is $\{Q,S\}$. Checking the other options: (A) and (D) each include P or R, which fail, and (B) leaves out S, which works. The answer is (C) Q and S.