The image/reaction scheme is not visible in the prompt, so the exact structure of product $B$ cannot be reconstructed directly from the provided text alone. However, we can still infer the intended conclusion from the option patterns and standard reactions from the chapter aldehydes, ketones and carboxylic acids.

Step 1: Analyze the nature of the options

The options describe product $B$ in terms of:

1. Optical activity / racemic nature
2. Acidic or neutral character
3. Reaction with bromine / NaHCO$_3$

A compound that gives a gas with saturated NaHCO$_3$ must be a carboxylic acid, because:

So if option C is true, product $B$ must contain a carboxylic acid group.

Step 2: Racemic mixture vs optically active

If a product is formed through addition to a planar carbonyl group or via creation of a new chiral center without stereocontrol, the product is generally obtained as a racemic mixture, not optically active.

Thus:

• “optically active” is usually incorrect unless the reaction is stereospecific or starts from a chiral substrate.
• “racemic mixture” is the more likely description in such standard JEE organic problems.

Step 3: Check the options conceptually

Option A:

racemic mixture and is neutral

• If product is neutral, it is not a carboxylic acid.
• This conflicts with the common reaction patterns where hydrolysis/oxidation products in this chapter often give acids.

Option B:

optically active and adds one mole of bromine

• “Adds one mole of bromine” suggests a C=C bond.
• Also says optically active, which is less likely in a standard non-stereoselective synthesis.
• Unlikely.

Option C:

racemic mixture and gives a gas with saturated NaHCO$_3$ solution

• Gas evolution with NaHCO$_3$ strongly indicates a carboxylic acid.
• Racemic mixture is consistent if a new chiral center is formed in a non-stereoselective reaction.
• This is chemically very plausible.

Option D:

optically active alcohol and is neutral

• Neutral alcohol would not react with NaHCO$_3$.
• “Optically active” again is less likely without stereochemical control.
• Unlikely.

Step 4: Conclusion

The only option that matches standard organic behavior well is:

This means product $B$ is most likely a chiral carboxylic acid formed as a racemic mixture.