GATE EC 2022 Set 1 — Question 10
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General Aptitude → Spatial Aptitude → Transformation of Shapes (Translation, Rotation, Scaling, Mirroring)
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Question
Consider the following square with the four corners and the center marked as and respectively.
Let and represent the following operations:: rotation of the square by degree with respect to the axis.
: rotation of the square by degree with respect to the axis.
: rotation of the square by degree clockwise with respect to the axis perpendicular, going into the screen and passing through the point .Consider the following three distinct sequences of operation (which are applied in the left to right order).(1)
(2)
(3) Which one of the following statements is correct as per the information provided above?
: rotation of the square by degree with respect to the axis.
: rotation of the square by degree clockwise with respect to the axis perpendicular, going into the screen and passing through the point .Consider the following three distinct sequences of operation (which are applied in the left to right order).(1)
(2)
(3) Which one of the following statements is correct as per the information provided above?
Correct answer
(B) The sequence of operations (1) and (3) are equivalent
Solution
Let the vertices of the square be represented as a sequence .
This result corresponds to a rotation of the original square about its center .
1.Operation : Rotation by about the axis. This is a reflection across the diagonal . Vertices and remain fixed, while and are swapped.
2.Operation : Rotation by about the axis. This is a reflection across the diagonal . Vertices and remain fixed, while and are swapped.
3.Combined Operation : Applying then :
This result corresponds to a rotation of the original square about its center .
4.Operation : A clockwise rotation about . Thus, is a rotation, and is a rotation (the identity operation ).
From step 3, we see that .5.Evaluating the Sequences:
- Sequence (1): (Identity)
- Sequence (2): ( rotation)
- Sequence (3): (Identity)
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