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GATE 2022 MT – Question 10

General Aptitude · Spatial Aptitude: Spatial Reasoning and Visualization · 2 marks · Multiple choice

A square has corners P, Q, R, S and centre T as shown. Define X as a 180° rotation about the S–Q axis, Y as a 180° rotation about P–R, and Z as a 90° clockwise rotation about the axis perpendicular to the screen, directed into it through T.

Compare sequences applied left to right: (1) XYZZ; (2) XY; (3) ZZZZ. Which statement is correct?

Diagram for GATE 2022 MT question 10
  1. The sequence of operations (1) and (2) are equivalent
  2. The sequence of operations (1) and (3) are equivalent
  3. The sequence of operations (2) and (3) are equivalent
  4. The sequence of operations (1), (2) and (3) are equivalent

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Correct answer: (B) The sequence of operations (1) and (3) are equivalent

Explanation

The two 180° rotations about perpendicular in-plane diagonals compose to a 180° rotation about the square’s normal: $XY=Z^2$.

Thus $XYZZ=Z^4$ is the identity, as is sequence 3. Sequence 2 alone is not the identity. Correct comparison: **1 and 3, B**.