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
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
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
Options
Answer: (2) 24
Explanation:
Step 1: Define variables Let: x = number of correct answers y = number of wrong answers z = number of unattempted questions Given: Total questions: x + y + z = 75 Scoring scheme: Correct: +3 marks Wrong: -1 mark Unattempted: +1 mark Total marks scored: 3*x - y + z = 97 Condition: unattempted questions more than attempted: z > x + y Step 2: Express y in terms of x and z y = 75 - x - z Substitute into total marks: 3*x - (75 - x - z) + z = 97 Simplify: 3x - 75 + x + z + z = 97 4x + 2z - 75 = 97 4x + 2z = 172 2x + z = 86 (Equation 1) Step 3: Condition on unattempted questions Attempted questions = x + y = x + (75 - x - z) = 75 - z Given z > x + y → z > 75 - z → 2*z > 75 → z > 37.5 Since z is integer: z ≥ 38 Step 4: Express x in terms of z From Equation 1: 2x + z = 86 → 2x = 86 - z → x = (86 - z)/2 x must be integer → 86 - z must be even Minimum z = 38 → 86 - 38 = 48 → x = 24 Step 5: Find y y = 75 - x - z = 75 - 24 - 38 = 13 Check condition: z > x + y → 38 > 24 + 13 = 37 ✅ Step 6: Verify total marks Total marks = 3x - y + z = 324 - 13 + 38 = 72 - 13 + 38 = 97 ✅ Answer: Maximum number of correct answers Rayan could have given = 24
Explanation:
Step 1: Define variables Let: x = number of correct answers y = number of wrong answers z = number of unattempted questions Given: Total questions: x + y + z = 75 Scoring scheme: Correct: +3 marks Wrong: -1 mark Unattempted: +1 mark Total marks scored: 3*x - y + z = 97 Condition: unattempted questions more than attempted: z > x + y Step 2: Express y in terms of x and z y = 75 - x - z Substitute into total marks: 3*x - (75 - x - z) + z = 97 Simplify: 3x - 75 + x + z + z = 97 4x + 2z - 75 = 97 4x + 2z = 172 2x + z = 86 (Equation 1) Step 3: Condition on unattempted questions Attempted questions = x + y = x + (75 - x - z) = 75 - z Given z > x + y → z > 75 - z → 2*z > 75 → z > 37.5 Since z is integer: z ≥ 38 Step 4: Express x in terms of z From Equation 1: 2x + z = 86 → 2x = 86 - z → x = (86 - z)/2 x must be integer → 86 - z must be even Minimum z = 38 → 86 - 38 = 48 → x = 24 Step 5: Find y y = 75 - x - z = 75 - 24 - 38 = 13 Check condition: z > x + y → 38 > 24 + 13 = 37 ✅ Step 6: Verify total marks Total marks = 3x - y + z = 324 - 13 + 38 = 72 - 13 + 38 = 97 ✅ Answer: Maximum number of correct answers Rayan could have given = 24
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