Step 1: Understanding the Concept:
Identifying important horticultural vegetable hybrids, their origin, and crop classification is necessary for professional horticultural exams.
Step 2: Detailed Explanation:
- Statement (I) is true: 'Kashi Bhairav' (VRB-10) is indeed a high-yielding $F_1$ hybrid of brinjal (eggplant, Solanum melongena L.) developed by the ICAR-Indian Institute of Vegetable Research (IIVR), Varanasi. It is known for its long, purple fruits and resistance to major bacterial wilt.
- Statement (II) is false: 'Shitla Jyoti' (often spelled Sheetla Jyoti) is not a hybrid of watermelon. It is a popular $F_1$ hybrid variety of cabbage (Brassica oleracea var. capitata) characterized by round, compact heads with high transport resistance.
Therefore, Statement (I) is true but Statement (II) is false.
Step 3: Final Answer:
Statement (I) is true, and Statement (II) is false.