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

Given:
- Total questions = \(75\) 
- Correct answer: +3 marks
- Wrong answer: -1 mark
- Unattempted question: +1 mark

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

From the problem, we have:
\(C + W + U = 75 \quad \text{(Equation 1)}\)
\(3C - W + U = 97 \quad \text{(Equation 2)}\)
Also given: \(U > C + W\)

Step 1: From Equation 1, express \(U\)
\(U = 75 - C - W\)

Step 2: Substitute into Equation 2
\(3C - W + (75 - C - W) = 97\)
\((3C - C) - 2W + 75 = 97\)
\(2C - 2W = 22\)
\(C - W = 11 \quad \text{(Equation 3)}\)

Step 3: Use the inequality
Given: \(U > C + W\)
From Equation 1: \(U = 75 - C - W\)
So: \(75 - C - W > C + W\)
\(75 > 2C + 2W\)
\(37.5 > C + W\)
\(C + W < 38\)  (Equation 4)

Step 4: Solve using Equation 3 and Equation 4
From Equation 3: \(C = W + 11\)
Substitute into Equation 4:
\(C + W = (W + 11) + W = 2W + 11 < 38\)
\(2W < 27 \Rightarrow W < 13.5\)
So \(W \leq 13\)

Now \(C = W + 11 \Rightarrow \text{maximum C when W is maximum}\)
If \(W = 13\), then \(C = 24\)

Final Answer:
The maximum number of correct answers Rayan could have given is \(24\).
Correct Option: 24

Answer 2

Let: 
\(x\) = number of correct answers
\(y\) = number of incorrect answers
\(z\) = number of Un attempted questions

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

Step 1: Subtract equation (i) from (ii):
\((3x - y + z) - (x + y + z) = 97 - 75\)
\(2x - 2y = 22 \Rightarrow x - y = 11 \quad \text{(iii)}\)

Step 2: Add equation (i) and (ii):
\((x + y + z) + (3x - y + z) = 75 + 97\)
\(4x + 2z = 172 \Rightarrow 2x + z = 86 \quad \text{(iv)}\)

Step 3: Use the condition:
Unattempted questions > Attempted questions
\(z > x + y\)
From (i): \(x + y = 75 - z\)
So, \(z > 75 - z \Rightarrow 2z > 75 \Rightarrow z > 37.5\)
Since \(z\) must be an integer, minimum possible value of \(z = 38\)

Step 4: Substitute \(z = 38\) into equation (iv):
\(2x + 38 = 86 \Rightarrow 2x = 48 \Rightarrow x = 24\)

Final Answer:
The maximum number of correct questions solved is \(24\).