To solve this problem, we need to determine the number of questions that the student answered wrongly, given certain conditions.
Let's denote the following variables:
The total number of questions is 50, so we have:
\(x + y + z = 50\)
For each correct answer, the student scores 1 mark. For each wrong answer, the student loses \(\frac{1}{3}\) of a mark. For each question not attempted, the student loses \(\frac{1}{6}\) of a mark.
The net score of the student is given as 32. Thus, we can formulate the score equation as:
\(x - \frac{1}{3}y - \frac{1}{6}z = 32\)
We now have two equations:
To simplify and solve these equations, let's multiply Equation 2 by 6 to eliminate the fractions:
\(6x - 2y - z = 192\) (Equation 3)
Now, we'll subtract Equation 1 from Equation 3:
\((6x - 2y - z) - (x + y + z) = 192 - 50\)
This simplifies to:
\(5x - 3y - 2z = 142\) (Equation 4)
From Equation 1, we can express \(z\) as:
\(z = 50 - x - y\) (Equation 5)
Substitute Equation 5 into Equation 4:
\(5x - 3y - 2(50 - x - y) = 142\)
Simplify to get:
\(5x - 3y - 100 + 2x + 2y = 142\)
Which further simplifies to:
\(7x - y = 242\) (Equation 6)
From Equation 6, we can express \(y\) in terms of \(x\):
\(y = 7x - 242\)
Now, ensure \(y\) is a non-negative value:
Solving \(7x - 242 \geq 0\), we get \(x \geq \frac{242}{7} = 34.57\).
Thus, \(x\) must be at least 35 because \(x\) is an integer.
Substitute \(x = 35\) back in:
The minimum number of questions answered wrongly is 3.
Therefore, the correct answer is 3.