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The correct option is option-1, which states that statement 1 only is correct. Let`s dissect each statement to understand why.
Statement 1: The PSLV (Polar Satellite Launch Vehicle) that launched Cartosat-I and Hamsat satellites was a three-stage vehicle. This statement is correct. The PSLV is a four-stage vehicle, consisting of a solid-fueled first stage, a liquid-fueled second stage, a solid-fueled third stage, and a liquid-fueled fourth stage. Therefore, the PSLV used in this particular launch indeed had three stages.
Statement 2: The Cartosat-I and Hamsat satellites were separated from the PSLV exactly at the same time. This statement is not mentioned in the question and cannot be inferred from the information provided. Hence, statement 2 is incorrect.
In conclusion, only statement 1 is correct, and statement 2 is incorrect. Therefore, the correct answer is option-1.