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GATE 2022 EC – Question 10

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

Consider the following square with the four corners and the center marked as P, Q, R, S and T respectively.

Let X, Y and Z represent the following operations:

X: rotation of the square by 180 degree with respect to the S-Q axis.

Y: rotation of the square by 180 degree with respect to the P-R axis.

Z: rotation of the square by 90 degree clockwise with respect to the axis perpendicular, going into the screen and passing through the point T.

Consider the following three distinct sequences of operation (which are applied in the left to right order).

(1) XYZZ

(2) XY

(3) ZZZZ

Which one of the following statements is correct as per the information provided above?

Diagram for GATE 2022 EC 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

X and Y are flips about the two perpendicular diagonals; doing both gives a $180^\circ$ rotation about T. Z twice is also a $180^\circ$ rotation, so XYZZ is a $360^\circ$ rotation, i.e. the identity. ZZZZ is $4\times90^\circ=360^\circ$, also the identity. XY alone is a $180^\circ$ rotation, so only (1) and (3) are equivalent.