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?

EXPLANATION

The correct option is B - 81.

[as per provisional answerkey]

Solution

Let the number of correct answers be x and the number of wrong answers be y. Since the student attempted all questions, the total number of questions is ( $x + y$ ).

Step 1: Formulate equations based on the given conditions.
Case 1: 5 marks for correct, 2 marks deducted for wrong. Score = 69.
$5 x - 2 y = 69$ - (Equation 1)

Case 2: 4 marks for correct, 1 mark deducted for wrong. Score = 84.
$4 x - 1 y = 84$ - (Equation 2)

Step 2: Solve the system of linear equations.
From Equation 2, we can express y in terms of x:
$y = 4 x - 84$

Substitute this value of y into Equation 1:
$5 x - 2 (4 x - 84) = 69$
$5 x - 8 x + 168 = 69$
$- 3 x = 69 - 168$
$- 3 x = - 99$
$x = 33$

Now, find the value of y using $x = 33$ :
$y = 4 (33) - 84$
$y = 132 - 84$
$y = 48$

Step 3: Calculate the total number of questions.
Total questions = $x + y$
Total questions = $33 + 48 = 81$ .

Why the other options are incorrect

Option (a) - 99: This value does not satisfy the simultaneous equations; if total questions were 99, the scores would not align with the 69/84 results given.
Option (c) - 84: This is the score mentioned in the second scenario, not the total number of questions.
Option (d) - 79: This value results from a calculation error or incorrect substitution in the linear equations.

Key Concept

Solving a system of linear equations in two variables to determine unknown quantities based on multiple scoring scenarios.