Examine the following three pairs of possible isomers. Now state whether the pairs represent identical compounds or different isomers.

$(i)$ Number of geometrical isomers for the given structure $X$:
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$(ii)$ Number of cyclic structural isomers with molecular formula $C_5H_{10} = Y$
$(iii)$ Number of ethers with molecular formula $C_4H_{10}O = Z$
Find the value of $X + Y + Z$.

Determine the absolute configuration of the given molecule:

Which of the following pairs represent a pair of diastereomers?

$P$ and $Q$ are isomers of dicarboxylic acid $C_4H_4O_4$. Both decolorize $Br_2/H_2O$. On heating,$P$ forms a cyclic anhydride.
Upon treatment with dilute alkaline $KMnO_4$,$P$ as well as $Q$ could produce one or more than one from $S$,$T$,and $U$.
$1.$ Compounds formed from $P$ and $Q$ are,respectively:
$(A)$ Optically active $S$ and optically active pair $(T, U)$
$(B)$ Optically inactive $S$ and optically inactive pair $(T, U)$
$(C)$ Optically active pair $(T, U)$ and optically active $S$
$(D)$ Optically inactive pair $(T, U)$ and optically inactive $S$
$2.$ In the following reaction sequences,$V$ and $W$ are respectively:
Give the answer for question $1$ and $2$.

The circle represents the most basic atom in these molecules. Which of the following is the correct representation?