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
24
Step-by-Step Solution
Key idea: this is a linear-constraint scoring question. The trigger is that three types of questions exist: correct, wrong, and unattempted, each with a different mark. We need a count equation and a value equation.
Step 1: Define variables.
Let be correct answers, wrong answers, and unattempted questions.
Step 2: Write the count equation.
Total questions = 75, so:
.
Step 3: Write the score equation.
Correct gives +3, wrong gives -1, unattempted gives +1.
So:
.
Step 4: Use the condition on unattempted questions.
Attempted questions = .
Unattempted is higher than attempted:
.
Since , let attempted . Then .
The condition becomes:
.
Since is an integer, .
Step 5: Eliminate from the score equation.
From the count equation, .
Substitute into the score equation:
.
Step 6: Maximise under the attempted limit.
From , we get .
Attempted questions:
.
Since attempted must be at most 37:
.
To maximise , take the largest possible , which is 13.
Then .
Step 7: Verify.
If , , then attempted = 37 and .
Score = .
Also, unattempted 38 is higher than attempted 37.
Answer: 24.