The capital represents the state political level.
The Landtag works in Düsseldorf on the Rhine. Its members pass state laws and scrutinise the state government. The state ministries also have their offices in Düsseldorf, explaining the city’s political importance for the whole federal state.
Düsseldorf is also a city with its own municipal administration. The city administration and state government are therefore separate institutions, despite working in the same place. The mayor leads the city; the Minister President leads the state government.
The other three answers
Being well known and being the capital are different things.
Köln
Cologne is not North Rhine-Westphalia’s state capital. The question concerns a political function, rather than choosing the best-known city.
Bonn
Bonn was the capital of the former West German Federal Republic and is now designated a federal city. That federal role must be distinguished from Düsseldorf’s role as state capital.
Dortmund
Dortmund is not a state capital. NRW’s Landtag and state government work in Düsseldorf, so Dortmund does not match the role asked about.
Keep state capital, federal city and county administration separate.
A federal state can have many important cities without having several state capitals. The four answer names are not a vote about city size, football or cultural importance. The question concerns the official state-capital role.
Siegburg, as the seat of Rhein-Sieg-Kreis’s administration, is another level again. A county administration serves its county; state politics concerns all of North Rhine-Westphalia. Check a particular authority’s current address before visiting.
State capital since 1946 · City of Düsseldorf ↗North Rhine-Westphalia → Düsseldorf.
Bonn belongs to federal history; Düsseldorf is the required state capital.
Three words for this question
- die Landeshauptstadtstate capital
- The capital of a German federal state.
- die Landesregierungstate government
- The government for the whole federal state, distinct from a city administration.
- die Bundesstadtfederal city
- Bonn’s special designation reflecting its role at federal level.