For this question you will need to look at the opposite of what you would look for in finding the most acidic aniline. A more detailed explanation will follow anyway.
The most basic aniline is the one in which electron density is supplied to it from one of the three substituents (methyl, nitro, methoxy).
Since an amino group's basic character is determined largely by its lone pairs, and these lone pairs can be delocalized into the ring, you will need a substituent that will "re-supply" this lone pair through resonance.
Alkyl groups cannot donate electrons through resonance (review the conditions required for electron delocalization).
You've got the reverse thinking. You're not trying to find which substituent "feels" withdrawal more by the amino group, but how the amino group "feels" electron density by each substituent.
Inductive effects are too weak to really matter compared to resonance effects. Since alkyl groups cannot participate through electron resonance donation, we can rule out methyl.
Nitro groups, as you know, deactivate benzene rings. If a nitro group is placed
para to NH
2, the lone pairs of NH
2 can be delocalized onto it. So you know that a nitro group would increase
acidity, not basicity.
Basicity will follow the opposite trend to acidity. So find the most activating group for the most basic aniline.
Think of it this way. You know that in order to have a strong acid, its conjugate base must be stable (much weaker), and vice versa (think of water auto-ionizing; 2H
2O --> HO
- + H
3O
+. HO is a stonger base because water is a weaker acid).
So first, protonate the NH
2, giving NH
3+, it's conjugate acid. You know that any time you protonate something, you increase its acidity (protonating water gives H
3O
+, a much stronger acid than water).
So we've now set down the following rule: the stronger the base, the more stable its conjugate acidic form. We've just increased the Ka of NH
2 by protonating it, so now we must find which substituent (methyl, methoxy, or nitro) will stabilize the acid the best, giving the stonger conjugate base (NH
2).
Placing MeO
para to NH
3+, see what happens when you move a lone pair of the O of MeO into the ring. The C atom connecting the protonated amino group has a negative charge (an extra lone pair), and it is this extra lone pair that can conjugate with the positively charged N to stabilize its charge.
More stable acidic form, the stronger is its basic form.
Now the question you must ask yourself is, can a nitro group do this?