GATE 2022 EE GA Q.9: solution and answer

Official answer: ASource: Official answer key, GATE 2022 EE (IIT Kharagpur)

1. Number the hexagon's vertices $1,2,\dots ,6$ in order around it. A segment joining two vertices has one of three lengths: a side (adjacent vertices, length $a$), a short diagonal (one vertex skipped, $\sqrt{3}a$) or a long diagonal (opposite vertices, $2a$).

2. Direction rule for a regular hexagon: the segment joining vertices $i$ and $j$ has a direction fixed by $(i+j)\mathrm{mod}6$. Two segments with the same value are parallel. Two segments whose values differ by $3$ are perpendicular.

3. Statement 1 says RS is longer than PQ, so the pair (PQ, RS) is (side, short diagonal), (side, long diagonal) or (short diagonal, long diagonal).

4. Side with long diagonal: a side $i\text{-}(i+1)$ has an odd sum, e.g. $1\text{-}2\to 3$. A long diagonal $i\text{-}(i+3)$ has sum $2i+3$, which is also odd. Two odd sums cannot differ by $3$ (mod 6), so a side is never perpendicular to a long diagonal. Ruled out.

5. Side with short diagonal: by symmetry take the side $1\text{-}2$ (sum $3$). It is perpendicular only to the chords with sum $\equiv 0$, namely $1\text{-}5$ and $2\text{-}4$. Both share a vertex with the side, but R, S, P and Q are four different vertices. Ruled out.

6. So PQ is a short diagonal and RS a long diagonal. By symmetry take $\{P,Q\}=\{1,3\}$ (sum $4$). The perpendicular chords have sum $\equiv 1$, i.e. $1\text{-}6$, $2\text{-}5$, $3\text{-}4$; the only long diagonal among them is $2\text{-}5$. So $\{R,S\}=\{2,5\}$ and $\{T,U\}=\{4,6\}$.

7. Statement 3 (RU $\parallel$ TQ, so equal sums mod 6): checking the eight choices of $R\in \{2,5\}$, $U\in \{4,6\}$, $Q\in \{1,3\}$ leaves only two arrangements: $R=5,U=4,T=6,Q=3$ (both sums $9\equiv 3$), or $R=5,U=6,T=4,Q=1$ (sums $11$ and $5$, both $\equiv 5$). Going around the hexagon, these orders are P, S, Q, U, R, T and Q, S, P, T, R, U. Each is the mirror image of the other.

8. Take the first arrangement ($P=1,S=2,Q=3,U=4,R=5,T=6$). (A) RT $=5\text{-}6$ (sum $11\equiv 5$) and QS $=3\text{-}2$ (sum $5$): parallel, because these are opposite sides. True.

9. Check the others: (B) TQ $=6\text{-}3$ (sum $\equiv 3$) and PU $=1\text{-}4$ (sum $5$) are two different long diagonals, which cross at the centre, so they are not parallel. (C) RP sum $6\equiv 0$ and UQ sum $7\equiv 1$: they differ by $1$, not $3$, so not perpendicular. (D) QS sum $5$ and RP sum $0$: they differ by $1$, so not perpendicular. The mirror arrangement gives the same results. Only (A) holds.