Benzene on reaction with $A$ forms acetophenone,which on reaction with $B$ forms ethylbenzene. $A$ and $B$ are:

Complete reaction of acetaldehyde with excess formaldehyde,upon heating with conc. $NaOH$ solution,gives $P$ and $Q$. Compound $P$ does not give Tollens' test,whereas $Q$ on acidification gives positive Tollens' test. Treatment of $P$ with excess cyclohexanone in the presence of catalytic amount of $p$-toluenesulfonic acid $(PTSA)$ gives product $R$. Sum of the number of methylene groups $(-CH_2-)$ and oxygen atoms in $R$ is....

Identify compound $A$ from the following reaction:
$A + C_2H_5MgBr$ $\xrightarrow[ether]{dry} B$ $\xrightarrow{H_3O^{+}} 3-\text{methylpentan}-3-\text{ol}$

The compound $(X)$ in the given reaction is:

The correct order of rate of reaction toward nucleophilic addition reaction for the following compounds is:
$(a)$ $p-NO_2-C_6H_4-CHO$
$(b)$ $C_6H_5-CHO$
$(c)$ $p-OMe-C_6H_4-CHO$
$(d)$ $CH_3-CO-CH_2CH_3$
$(e)$ $C_6H_5-CO-C_6H_5$

The major product $X$ formed in the following reaction is: