Step 1: Analyze the scoring system:
- Correct answers contribute +3 marks.
- Wrong answers deduct 1 mark (-1 mark).
- Unattempted questions award +1 mark.
Step 2: Define variables representing quantities:
Let:
- \(\mathbf{C}\): The count of correct responses.
- \(\mathbf{W}\): The count of incorrect responses.
- \(\mathbf{U}\): The count of unattempted questions.
Step 3: Input data summary:
- \(\mathbf{C + W + U = 75}\) (Total number of questions)
- \(\mathbf{3C - W + U = 97}\) (Total marks achieved)
Step 4: Constraint on unattempted questions:
The number of unattempted questions (\(\mathbf{U}\)) exceeds the number of questions answered (\(\mathbf{C + W}\)). Therefore, \(\mathbf{U > C + W}\).
Step 5: Isolate U in terms of C:
Using the provided equations, we can express \(\mathbf{U}\) in relation to \(\mathbf{C}\):
\(\mathbf{U = 97 - 3C + W}\)
Step 6: Substitute U into the inequality and simplify:
\(\mathbf{97 - 3C + W > C + W}\)
Simplification yields:
\(\mathbf{97 - 3C > C}\)
Rearranging the inequality:
\(\mathbf{97 > 4C}\)
Dividing by 4:
\(\mathbf{C < 24.25}\)
Step 7: Determine the maximum integer value for C:
As \(\mathbf{C}\) must be an integer (representing the count of correct answers), the highest possible value for \(\mathbf{C}\) is 24.
Final Answer:
The maximum possible number of correct answers Rayan could have achieved is 24.