Q.10 An ant walks in a straight line on a plane leaving behind a trace of its
movement. The initial position of the ant is at point P facing east.
The ant first turns 72o anticlockwise at P, and then does the following two
steps in sequence exactly FIVE times before halting.
1. moves forward for 10 cm.
2. turns 144o clockwise.
The pattern made by the trace left behind by the ant is
Ant starts at point P facing east, turns 27° anticlockwise, then repeats (forward 10 cm + 144° clockwise turn) exactly twice, creating a specific geometric trace pattern among the given options.
Initial Setup
The ant begins at P facing east (0° from positive x-axis). A 27° anticlockwise turn shifts its direction to 27° north of east (bearing 27°). This aligns with standard geometry problems where directions accumulate via vector additions.
Step-by-Step Path Calculation
After the initial 27° left turn, the sequence executes twice:
- First iteration: Moves 10 cm at 27°, reaches Q. Position: Qx = 10cos(27° ) ≈ 8.94 cm east, Qy = 10sin(27°) ≈ 4.55 cm north. Then turns 144° clockwise to new direction: 27° + 144° = 171° (nearly west).
- Second iteration: Moves 10 cm at 171°, reaches R. Position from Q: Rx ≈ 8.94 – 10cos(9°) ≈ 8.94 – 9.90 ≈ -0.96 cm (westward closure), Ry ≈ 4.55 + 10sin(9°) ≈ 4.55 + 1.56 ≈ 6.11 cm north. Final 144° clockwise turn is irrelevant as it halts.
The path PQ ≈ 10 cm straight initially, QR closes near P but offset north, forming a narrow V or incomplete triangle, not matching closed polygons.
Option Analysis
| Option | Description | Why Incorrect |
|---|---|---|
| (A) Pentagon | Requires 5 equal 10 cm sides (PQ-QR-RS-ST-UP) | Actual path has only 2 moves (20 cm total), insufficient for 5 sides |
| (B) Hexagon | Needs 6 equal sides (PQ-QR-RS-ST-UP, implied closure) | Only 2 segments drawn; no 60° internals for regular hexagon |
| (C) Star (left turns) | Labels suggest 5-point star (SQ-QT-RP-RS, 3 cm sides mismatch 10 cm) | Directions don’t yield intersecting star paths |
| (D) Star (right turns) | 8 segments labeled 10 cm each | Sequence produces only 2 traces. Mismatch in side count and no full closure/star intersections |
Key Insight: No option perfectly matches the open 2-segment trace (PQ-R, ~120° angle), but closest is (A) if misinterpreting as partial pentagon start. Question likely tests direction geometry for CSIR NET spatial reasoning.
Problem Breakdown
Initial east (0°) becomes 27° post-turn. First leg: 10 cm at 27°. Second: 144° clockwise to 171°, another 10 cm. Net displacement ~6 cm north, minimal east-west closure—no full polygon.
Why Not Closed Shapes?
Pentagon/hexagon/star require 5/6/5+ sides and 360°/n internals. Here, 2×144°=288° turn total +27° yields irregular V-path, not symmetric. Similar to GATE variants (72°+144°×5=star).
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