Question:medium

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

Updated On: Jan 15, 2026
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Correct Answer: 24

Solution and Explanation

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.

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