Explanation:

- Noble's Helen: This is a butterfly found in the Indian subcontinent, known for its elegant appearance but is not associated with Jammu and Kashmir as their official butterfly.

- Kaisar-i-Hind: Known for its large, striking appearance, particularly in northeastern India, but not related to Jammu and Kashmir's official butterfly status.

- Blue Duke: A butterfly found in the Himalayan ranges, recently named as the official butterfly of Jammu and Kashmir.

- Blue Pansy: This butterfly is widespread in various parts of India and Asia but is not the official butterfly of Jammu and Kashmir.

Blue Duke is the correct answer for the official butterfly of Jammu and Kashmir.