For which value(s) of `rho` will the lines represented by the following pair of linear equations be paralle
`3x - y- 5=0`
`6x- 2y - p=0`

Options

A. all real values except 10
B. 10
C. `5//2`
D. `1//2`

Explanation

To determine the value(s) of \( p \) for which the lines represented by the equations \( 3x - y - 5 = 0 \) and \( 6x - 2y - p = 0 \) are parallel, we can follow these steps:

### Step 1: Identify the coefficients
The first equation can be rewritten in the standard form \( Ax + By + C = 0 \):
- For the first equation \( 3x - y - 5 = 0 \):
- \( A_1 = 3 \)
- \( B_1 = -1 \)
- \( C_1 = -5 \)

The second equation is:
- For the second equation \( 6x - 2y - p = 0 \):
- \( A_2 = 6 \)
- \( B_2 = -2 \)
- \( C_2 = -p \)

### Step 2: Use the condition for parallel lines
For two lines to be parallel, the following condition must hold:
\[ \frac{A_1}{A_2} = \frac{B_1}{B_2} \quad \text{and} \quad \frac{A_1}{A_2} \neq \frac{C_1}{C_2} \]

### Step 3: Calculate the ratios
First, we calculate \( \frac{A_1}{A_2} \) and \( \frac{B_1}{B_2} \):
\[ \frac{A_1}{A_2} = \frac{3}{6} = \frac{1}{2} \]
\[ \frac{B_1}{B_2} = \frac{-1}{-2} = \frac{1}{2} \]

Since both ratios are equal, we can conclude that the lines are parallel if the third condition is satisfied.

### Step 4: Set up the inequality for \( C \)
Now we need to ensure that:
\[ \frac{C_1}{C_2} \neq \frac{1}{2} \]
Substituting the values of \( C_1 \) and \( C_2 \):
\[ \frac{-5}{-p} \neq \frac{1}{2} \]
This simplifies to:
\[ \frac{5}{p} \neq \frac{1}{2} \]

### Step 5: Solve the inequality
Cross-multiplying gives:
\[ 5 \cdot 2 \neq p \implies 10 \neq p \]

### Conclusion
Thus, the lines represented by the equations will be parallel for all values of \( p \) except \( 10 \).

### Final Answer
The value(s) of \( p \) for which the lines are parallel is:
\[ p \in \mathbb{R} \setminus \{10\} \]