In an objective type question paper, 5 marks are awarded for a correct answer and 2 marks are deducted for a wrong answer. A student attempted all the questions and got a score of 69. Had he been awarded 4 marks for a correct answer and 1 mark deducted for a wrong answer, he would have scored 84. How many questions were there in the question paper?
- (a) 99
- (b) 81 ✓ UPSC's answer
- (c) 84
- (d) 79
Why the answer is (b)
• Let $x$ be the number of correct answers and $y$ be the number of wrong answers.
• The total number of questions is $x + y$.
• Based on the first scoring system (5 for correct, -2 for wrong), the equation is $5x - 2y = 69$.
• Based on the second scoring system (4 for correct, -1 for wrong), the equation is $4x - y = 84$.
• Multiply the second equation by 2 to get $8x - 2y = 168$.
• Subtract the first equation from this new equation: $(8x - 2y) - (5x - 2y) = 168 - 69$, which simplifies to $3x = 99$, so $x = 33$.
• Substitute $x = 33$ into $4x - y = 84$: $4(33) - y = 84 \Rightarrow 132 - y = 84 \Rightarrow y = 48$.
• The total number of questions is $x + y = 33 + 48 = 81$.
Why the other options are wrong
- (a) 99
- 99 is the value of $3x$ derived during the calculation, not the total number of questions.
- (c) 84
- 84 is the score obtained under the second hypothetical scoring system, not the total number of questions.
- (d) 79
- 79 is not a solution to the system of linear equations derived from the problem statement.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2026, held on 24 May 2026. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.