Question

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

Answer 1

Correct Answer: 24

Step 1: Break down the information given in the problem:

• Correct answers: +3 marks
• Wrong answers: -1 mark
• Unattempted questions: +1 mark

Step 2: Define variables:

Let:

• \(C\): Number of correct answers
• \(W\): Number of wrong answers
• \(U\): Number of unattempted questions

Step 3: We are given the following information:

• \(C + W + U = 75\) (total questions)
• \(3C - W + U = 97\) (total marks)

Step 4: The condition on unattempted questions:

We are told that the number of unattempted questions \(U\) is greater than the number of attempted questions \((C + W)\), so \(U > C + W\).

Step 5: Solve for the maximum value of C:

From the equations above, we can solve for \(U\) in terms of \(C\):

\(U = 97 - 3C + W\)

Step 6: Substituting \(U\) into the inequality:

\(97 - 3C + W > C + W\)

Now simplify:

\(97 - 3C > C\)

Subtracting \(C\) from both sides:

\(97 > 4C\)

Dividing by 4:

\(C < 24.25\)

Step 7: Conclusion:

Since the number of correct answers \((C)\) must be a whole number, the maximum possible value for \(C\) is 24.

Final Answer:

The maximum number of correct answers Rayan could have given in the examination is 24.

Answer 2

Let’s assume the following:

• \(x\) be the number of correct questions.
• \(y\) be the number of incorrect questions.
• \(z\) be the number of unattempted questions.

Given:

• \(x + y + z = 75 \ \ \ \text{(i)}\)
• \(3x - y + z = 97 \ \ \ \text{(ii)}\)

Step 1: Subtract Equation (i) from Equation (ii):

\((2) - (1) \Rightarrow x - y = 11\)

Step 2: Add Equation (i) and Equation (ii):

\((1) + (2) \Rightarrow 2x + z = 86\)

Step 3: Use the inequality for unattempted questions:

\(z > x + y\)

\(z > 75 - z\)

\(z > 37.5\)

Step 4: Solve for \(z\):

From the above equation, the minimum possible value of \(z\) is 38.

Step 5: Substitute the value of \(z\) into the equation:

\(2x + 38 = 86\)

\(2x = 48\)

\(x = 24\)

Final Answer:

Therefore, the maximum number of correct questions solved is 24.