Step 1: Interpret 2,4-DNP test.
2,4-DNP reagent gives red-orange precipitate with compounds having carbonyl group.
Carbonyl compounds include:
\[
\text{aldehydes and ketones}.
\]
So compound \(P\) must contain:
\[
>C=O
\]
Step 2: Interpret Fehling's test.
Fehling's reagent is reduced by aliphatic aldehydes.
The question says \(P\) does not reduce Fehling's reagent.
So \(P\) is not an aliphatic aldehyde.
It is most likely a ketone.
Step 3: Identify \(P\).
Among aromatic carbonyl compounds, acetophenone is:
\[
\ce{C6H5COCH3}
\]
It is an aromatic ketone.
It gives 2,4-DNP test but does not reduce Fehling's reagent.
So:
\[
P=\text{Acetophenone}.
\]
Step 4: Oxidation of acetophenone.
On strong oxidation, side chain of alkyl benzene or aryl ketone is converted into:
\[
\ce{-COOH}
\]
Thus, acetophenone gives:
\[
\ce{C6H5COOH}
\]
This is benzoic acid.
Step 5: Sodium bicarbonate test.
Carboxylic acids react with sodium bicarbonate to produce carbon dioxide gas.
\[
\ce{RCOOH + NaHCO3 -> RCOONa + CO2 + H2O}
\]
Effervescence confirms carboxylic acid.
Therefore:
\[
Q=\text{Benzoic acid}.
\]
Hence:
\[
P=\text{Acetophenone},\quad Q=\text{Benzoic acid}.
\]