The letters P, Q, R, S, T and U are to be placed one per vertex on a regular convex hexagon, but not necessarily in the same order. Consider the following statements: - The line segment joining R and S is longer than the line segment joining P and Q. - The line segment joining R and U is perpendicular to the line segment joining P and T. - The line segment joining R and U is parallel to the line segment joining Q and T. Which of the given statements is/are CORRECT? A. Statement 1 and Statement 2 B. Statement 2 only C. Statement 1 only D. Statement 3 only

GATE 2022 · General Aptitude · Statements Follow · medium

Answer: Statements 1 and 2 are correct for a valid placement of the six letters on the hexagon vertices. The answer is A.

  1. Set up hexagon coordinates: Place hexagon with centre at origin, radius 1. Vertices at angles 0, 60, 120, 180, 240, 300 degrees: V1=(1,0), V2=(0.5, sqrt(3)/2), V3=(-0.5, sqrt(3)/2), V4=(-1,0), V5=(-0.5,-sqrt(3)/2), V6=(0.5,-sqrt(3)/2). There are 6! = 720 placements; we analyse geometrically.
  2. Try placement: P=V1, Q=V2, R=V3, S=V4, T=V5, U=V6: P=(1,0), Q=(0.5,sqrt(3)/2), R=(-0.5,sqrt(3)/2), S=(-1,0), T=(-0.5,-sqrt(3)/2), U=(0.5,-sqrt(3)/2). Stmt 1: |RS| = dist(R,S) = dist(V3,V4)=1 (adjacent); |PQ|=dist(V1,V2)=1 (adjacent) => |RS|=|PQ|, NOT longer. Try other placement.
  3. Try placement: P=V1, Q=V2, R=V2... rethink — use opposite/skip-one: Try P=V1, Q=V2 (adjacent, dist=1), R=V3, S=V6 (skip-two: dist=sqrt(3)~1.73 > 1). Now dist(RS)=dist(V3,V6): V3=(-0.5,sqrt(3)/2), V6=(0.5,-sqrt(3)/2). dist=sqrt(1+3)=2 (opposite!). So dist(RS)=2 > dist(PQ)=1. Statement 1 is TRUE for this placement. Now check Statements 2 and 3 with T=V4, U=V5 (remaining vertices).
  4. Check Statement 2: RU perpendicular to PT: R=V3=(-0.5, sqrt(3)/2), U=V5=(-0.5,-sqrt(3)/2). Direction vector RU = (0,-sqrt(3)) — vertical. P=V1=(1,0), T=V4=(-1,0). Direction vector PT = (-2,0) — horizontal. Vertical perpendicular to horizontal => RU perp PT. Statement 2 is TRUE.
  5. Check Statement 3: RU parallel to QT: RU direction = (0,-sqrt(3)) (vertical). Q=V2=(0.5,sqrt(3)/2), T=V4=(-1,0). Direction QT = (-1-0.5, 0-sqrt(3)/2) = (-1.5, -sqrt(3)/2). This is NOT vertical (has non-zero x-component). So RU is NOT parallel to QT. Statement 3 is FALSE.
  6. Determine the answer: For the placement P=V1,Q=V2,R=V3,S=V6,T=V4,U=V5: Statement 1 TRUE (|RS|=2 > |PQ|=1), Statement 2 TRUE (RU vertical, PT horizontal => perpendicular), Statement 3 FALSE (RU vertical, QT non-vertical => not parallel). So Statement 1 AND Statement 2 are correct. Answer is A.