Step 1: Note the two stereospecific rules.
Bromine in \(CCl_4\) adds anti across a double bond (through a bromonium ion), while cold alkaline \(KMnO_4\) adds the two \(-OH\) groups syn (same face). These two facts decide every product here.
Step 2: Predict Statement I product.
trans-but-2-ene with anti addition of \(Br_2\) gives the meso form of 2,3-dibromobutane. The meso product is the only correct outcome.
Step 3: Check the figure in Statement I.
The Newman projection drawn in Statement I does not show a centre of inversion or an internal mirror plane; it shows a chiral (d or l) isomer instead of the required meso form. So the structure shown is wrong, and Statement I is incorrect.
Step 4: Predict Statement II product.
cis-but-2-ene with syn addition of cold alkaline \(KMnO_4\) gives the meso form of butane-2,3-diol.
Step 5: Check the figure in Statement II.
Rotating the back carbon of the drawn projection lets every front group eclipse its identical back group, revealing an internal plane of symmetry. That is exactly meso-butane-2,3-diol, so Statement II is correct.
Step 6: Combine the verdicts.
Statement I is incorrect and Statement II is correct, which is option 1.
\[ \boxed{\text{Statement I is incorrect but Statement II is correct}} \]