Concept:
Calcium undergoes a sequence of reactions leading to calcium oxide, calcium hydroxide, calcium carbonate and finally calcium bicarbonate. Knowledge of common names and properties of these compounds is essential.
Step 1: Identify \(X\).
Calcium burns in oxygen to form calcium oxide.
\[
2Ca + O_2 \rightarrow 2CaO
\]
Hence,
\[
X=CaO
\]
which is known as quick lime.
Step 2: Identify \(Y\).
\[
CaO + H_2O \rightarrow Ca(OH)_2
\]
Therefore,
\[
Y=Ca(OH)_2
\]
which is slaked lime.
Step 3: Identify \(Z\).
\[
Ca(OH)_2 + CO_2 \rightarrow CaCO_3 + H_2O
\]
Hence,
\[
Z=CaCO_3
\]
which is calcium carbonate.
Step 4: Identify \(W\).
In excess carbon dioxide and water,
\[
CaCO_3 + CO_2 + H_2O
\rightarrow
Ca(HCO_3)_2
\]
Thus,
\[
W=Ca(HCO_3)_2
\]
which is soluble in water.
Step 5: Check the statements.
Statement I:
\[
Ca(HCO_3)_2
\]
is soluble in water.
Correct.
Statement II:
\[
CaCO_3
\]
is used as a mild abrasive in toothpaste.
Correct.
Statement III:
\(X\) is \(CaO\) (quick lime), not slaked lime.
Incorrect.
Therefore,
\[
\boxed{\text{I and II only}}
\]