In an examination, there were 75 questions. 3 marks were awarded for each correct answer, 1 mark was deducted for each wrong answer and 1 mark was awarded for each unattempted question. Rayan scored a total of 97 marks in the examination. If the number of unattempted questions was higher than the number of attempted questions, then the maximum number of correct answers that Rayan could have given in the examination is

Question

Accepted Answer

To solve this problem, we analyze the scoring system:

1. **Marks Allocation Rules:**  
   - Correct answer: +3 marks.  
   - Wrong answer: -1 mark.  
   - Unattempted question: +1 mark.

2. **Re-imagining the scoring system:**  
   - Every question is initially worth +1 mark (unattempted baseline).  
   - Correct answers add an additional $$ +2 $$ marks ($$ +3 - 1 $$).  
   - Wrong answers subtract $$ -2 $$ marks ($$ -1 - (+1) $$).

Therefore, Rayan starts with $$ 75 $$ marks (for all unattempted questions). The difference between his final score and $$ 75 $$ comes from his attempts:  
   $$
   	ext{Final Score} = 75 + (2 	imes 	ext{Correct Answers}) - (2 	imes 	ext{Wrong Answers})
   $$
   Rayan's final score is $$ 97 $$, so:  
   $$
   97 = 75 + (2R - 2W)
   $$
   Simplifying:  
   $$
   R - W = 11
   $$

3. **Unattempted Questions Condition:**  
   Let $$ R $$ be the number of correct answers, $$ W $$ be the number of wrong answers, and $$ U $$ be the number of unattempted questions. Then:  
   $$
   R + W + U = 75
   $$
   Since $$ U > R + W $$:  
   $$
   U > 75 - U
   $$
   Solving:  
   $$
   U > 37.5
   $$
   Thus, $$ U \geq 38 $$.

4. **Substituting for $$ U = 75 - R - W$$:**  
   From $$ R - W = 11$$, substitute $$ W = R - 11$$:  
   $$
   U = 75 - R - (R - 11)
   $$
   Simplify:  
   $$
   U = 86 - 2R
   $$
   Since $$ U > R + W$$:  
   Substituting $$ W = R - 11$$:  
   $$
   U > R + (R - 11)
   $$
   Simplify:  
   $$
   U > 2R - 11
   $$
   Combining $$ U = 86 - 2R$$:  
   $$
   86 - 2R > 2R - 11
   $$
   Solve for $$ R$$:  
   $$
   97 > 4R
   $$
   Thus, $$ R  R + W (38 > 37)$$, this satisfies all conditions.

Thus, the maximum number of correct answers is **24**.