Logical Deduction

1. 
   Analyze the Chemistry Data from the Table:
   

   We need to find the mark range that corresponds to at least 23 eligible students. Let's look at the key data points for Chemistry:

   
   
   
   
   • Number of students with marks "30 and above" = 21.
   
   
   • Number of students with marks "20 and above" = 66.
   
   
   
   


2. 
   Apply the Eligibility Condition:
   

   The condition is that "at least 23 students were eligible."

   
   
   
   
   • If the minimum qualifying mark was 30, only 21 students would be eligible. This does not meet the "at least 23" requirement.
   
   
   • To include at least 23 students, the minimum qualifying mark must be set lower than 30.
   
   
   
   


3. 
   Determine the Mark Range:
   

   When the cut-off is lowered from 30 to 20, the number of eligible students jumps from 21 to 66. Since 23 is between 21 and 66, the actual cut-off mark must lie somewhere in the range between 20 and 30.

   
   

Final Answer: The minimum qualifying marks would lie in the range of 20 to 30.