1. Identify products formed with one mole of $H_2$

But-2-yne is: $CH_3-C\equiv C-CH_3$

On catalytic hydrogenation with one mole of hydrogen:

• cis-addition gives cis-but-2-ene
• trans-addition / dissolving metal type reduction gives trans-but-2-ene

So let:

• $A =$ cis-but-2-ene
• $B =$ trans-but-2-ene

This is the standard comparison for properties of cis/trans alkenes.

2. Check statement A: “A is more soluble than B.”

Cis isomer has a net dipole moment, while trans isomer is nearly non-polar. Hence cis-but-2-ene is relatively more soluble in polar solvents than trans-but-2-ene.

So statement A is correct.

3. Check statement B: “The boiling point & melting point of A are higher and lower than B respectively.”

For cis/trans alkenes:

• cis isomer is more polar $\Rightarrow$ stronger intermolecular attraction $\Rightarrow$ higher boiling point
• trans isomer is more symmetrical $\Rightarrow$ packs better in crystal lattice $\Rightarrow$ higher melting point

Therefore:

• boiling point of $A$ (cis) is higher than $B$ (trans)
• melting point of $A$ is lower than $B$

So statement B is correct.

4. Check statement C: “A is more polar than B because dipole moment of A is zero.”

This statement is self-contradictory.

For but-2-ene:

• cis-but-2-ene has non-zero dipole moment
• trans-but-2-ene has dipole moment nearly zero due to cancellation

Thus $A$ is indeed more polar than $B$, but not because dipole moment of $A$ is zero.

So statement C is incorrect.

5. Check statement D: “$Br_2$ adds easily to B than A.”

Addition of $Br_2$ to alkene depends mainly on electron density of the double bond and steric effects. In this cis/trans pair, cis-2-butene generally reacts more readily than trans due to steric/electronic factors in bromonium ion formation.

Hence saying that bromine adds more easily to B (trans) than A (cis) is incorrect.

So statement D is incorrect.

6. Identify incorrect statements

Incorrect statements are: $C \text{ and } D$

Thus the correct option is: $\boxed{D}$

7. Comparison with stored answer

Stored correct answer: D

Our derived answer: D

They match.