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Consider the following square with the four corners and the center marked as $\text{P, Q, R, S}$ and $\text{T}$ respectively.

Let $\text{X, Y}$ and $\text{Z}$ represent the following operations:

$\text{X:}$ rotation of the square by $180$ degree with respect to the $\text{S – Q}$ axis.

$\text{Y:}$ rotation of the square by $180$ degree with respect to the $\text{P – R}$ axis.

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

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

1. $\text{XYZZ}$
2. $\text{XY}$
3. $\text{ZZZZ}$

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

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

Given that, the square:

The  $\text{X, Y}$ and $\text{Z}$ represent the following operations:

• ${\color{Magenta}{\text{X:}}}$ rotation of the square by $180$ degree with respect to the $\text{S – Q}$ axis.

• ${\color{Green}{\text{Y:}}}$ rotation of the square by $180$ degree with respect to the $\text{P – R}$ axis.

• ${\color{Orange}{\text{Z:}}}$ rotation of the square by $90$ degree clockwise with respect to the axis perpendicular, going into the screen and passing through the point $\text{T}.$

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

1. ${\color{Red}{\text{XYZZ}:}}$

1. ${\color{Blue}{\text{XY:}}}$

1. ${\color{Purple}{\text{ZZZZ:}}}$

$\therefore$ The sequence of operations $(1)$ and $(3)$ are equivalent.

Correct Answer $:\text{B}$

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