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?

This question tests the application of simultaneous linear equations, a core component of the UPSC CSAT Quantitative Aptitude syllabus grounded in standard NCERT algebra.

Why the correct option is correct: Let the number of correct answers be c and wrong answers be w. Since the student attempted all questions, the total number of questions is c + w.

The conditions provide two linear equations:

1. 5c - 2w = 69 (Actual scoring matrix)
2. 4c - w = 84 (Hypothetical scoring matrix)

From equation (2), we isolate w: w = 4c - 84

Substituting this into equation (1) gives: 5c - 2(4c - 84) = 69 5c - 8c + 168 = 69 -3c = -99 ⟹ c = 33

Substitute c = 33 into the equation for w: w = 4(33) - 84 = 132 - 84 = 48

The total number of questions is c + w = 33 + 48 = 81. Thus, Option B is correct.

Why the wrong options are wrong:

• Option A (99): Solving the system with c + w = 99 fails to produce the integers required to satisfy both scoring conditions.
• Option C (84): This is a deliberate distractor trap. It matches the hypothetical score provided in the prompt, not the total number of questions.
• Option D (79): This arbitrary total does not mathematically align with the unique solution (c+w=81) constrained by the provided scoring matrices.

Takeaway / Mnemonic: For double-condition word problems, remember DTS: Define variables (correct/wrong), Translate text into simultaneous equations, and Substitute to solve.