In which one of the following pairs the central atoms exhibit $$sp^2$$ hybridization ?

We need to find the pair where both central atoms exhibit $$sp^2$$ hybridization.

$$BF_3$$: B has 3 bond pairs, 0 lone pairs. Hybridization = $$sp^2$$. ✓

$$NO_2^-$$: N has 2 bond pairs + 1 lone pair = 3 electron groups. Hybridization = $$sp^2$$. ✓

$$H_2O$$: O has 2 bond pairs + 2 lone pairs = 4 electron groups. Hybridization = $$sp^3$$. ✗

$$NO_2$$: N has 2 bond pairs + 1 unpaired electron = 3 regions. Hybridization = $$sp^2$$. ✓

$$NH_2^-$$: N has 2 bond pairs + 2 lone pairs = 4 electron groups. Hybridization = $$sp^3$$. ✗

The pair where both are $$sp^2$$ is $$BF_3$$ and $$NO_2^-$$.

The correct answer is Option 2: $$BF_3$$ and $$NO_2^-$$.