The correct answer is D. $\(\mathbf{2}\)$.

Explanation
To solve this problem, we can translate the verbal descriptions into algebraic equations using the formula:

$$
\(\text { Dividend }=(\text { Divisor } \text { × } \text { Quotient })+\text { Remainder }\)
$$


Step 1: Set up the equations
1. From the first sentence: When positive integer $x$ is divided by 11 , the quotient is $y$ and the remainder is 4 .

$$
\(x=11 y+4-(\text { Equation } 1)\)
$$

2. From the second sentence: When $2 x$ is divided by 8 , the quotient is $3 y$ and the remainder is 2 .

$$
\(\begin{gathered}
2 x=8(3 y)+2 \\
2 x=24 y+2-(\text { Equation } 2)
\end{gathered}\)
$$

Step 2: Solve the system of equations
First, simplify Equation 2 by dividing the entire equation by 2 :

$$
\(x=12 y+1\)
$$


Now, since we have two different expressions for $x$, we can set them equal to each other:

$$
\(11 y+4=12 y+1\)
$$


Subtract $\(11 y\)$ from both sides:

$$
\(\begin{gathered}
4=y+1 \\
y=3
\end{gathered}\)
$$


Step 3: Find the value of $x$
Substitute $y=3$ back into Equation 1:

$$
\(\begin{gathered}
x=11(3)+4 \\
x=33+4 \\
x=37
\end{gathered}\)
$$

(Quick Check: $\(2 x=74.74 \div 8=9\)$ with a remainder of 2 . Since $\(3 y=3(3)=9\)$, the second condition is satisfied.)

Step 4: Calculate the final value
The question asks for the value of $\(13 y-x\)$ :

$$
\(\begin{gathered}
13(3)-37 \\
39-37=2
\end{gathered}\)
$$