1. Identify the isomeric dicarboxylic acids

Given molecular formula: $C_4H_4O_4$.

A dicarboxylic acid with this formula and which decolorises $Br_2/H_2O$ must contain a $C=C$ bond. So the two geometrical isomers are:

• Maleic acid: cis-butenedioic acid
• Fumaric acid: trans-butenedioic acid

Their structures are:

$\text{Maleic acid: } HOOC-CH=CH-COOH \quad (cis)$ $\text{Fumaric acid: } HOOC-CH=CH-COOH \quad (trans)$

2. Use the anhydride clue to identify P

On heating, P forms a cyclic anhydride.

This is a characteristic property of maleic acid because in the cis-form the two $-COOH$ groups are close enough to lose water intramolecularly and form a 5-membered cyclic anhydride.

Hence,

$P = \text{maleic acid}$ $Q = \text{fumaric acid}$

3. Reaction with dilute alkaline $KMnO_4$

Cold dilute alkaline $KMnO_4$ adds two $OH$ groups across the double bond giving vicinal diols.

So both maleic acid and fumaric acid give 2,3-dihydroxysuccinic acid (tartaric acid type products), but stereochemistry differs.

4. Product from P = maleic acid (cis alkene)

Syn addition of $OH$ across cis-butenedioic acid gives:

$HOOC-CH(OH)-CH(OH)-COOH$

For cis alkene + syn addition, the product is the meso tartaric acid form.

Meso tartaric acid is optically inactive due to internal plane of symmetry.

So product from P is:

$S = \text{meso tartaric acid (optically inactive)}$

5. Product from Q = fumaric acid (trans alkene)

Syn addition of $OH$ across trans-butenedioic acid gives:

$HOOC-CH(OH)-CH(OH)-COOH$

But now stereochemistry leads to the enantiomeric pair:

• $(2R,3R)$ tartaric acid
• $(2S,3S)$ tartaric acid

These are optically active individually, but together form an optically inactive racemic pair.

Since the options mention pair $(T,U)$, this corresponds to the enantiomeric pair. Thus the pair $(T,U)$ is considered optically inactive pair.

6. Match with options

• From P: optically inactive $S$
• From Q: optically inactive pair $(T,U)$

Therefore the correct option is:

$\boxed{\text{B}}$

7. Comparison with stored answer

Stored correct answer: B

My derived answer: B

So they agree.