To solve the problem, we need to find the value of \(2\lambda - \mu\) given the vertices of triangle \(A\), \(B\), and \(C\) and the condition that the median through \(A\) is equally inclined to the positive direction of the axes.
### Step-by-Step Solution
1. **Identify the Coordinates of Points**:
- \( A = (2, 3, 5) \)
- \( B = (-1, 3, 5) \)
- \( C = (\lambda, 5, \mu) \)
2. **Find the Midpoint \(D\) of Segment \(BC\)**:
The coordinates of point \(D\) (the midpoint of \(B\) and \(C\)) can be calculated as follows:
\[
D = \left( \frac{x_B + x_C}{2}, \frac{y_B + y_C}{2}, \frac{z_B + z_C}{2} \right)
\]
Substituting the coordinates of \(B\) and \(C\):
\[
D = \left( \frac{-1 + \lambda}{2}, \frac{3 + 5}{2}, \frac{5 + \mu}{2} \right) = \left( \frac{\lambda - 1}{2}, 4, \frac{5 + \mu}{2} \right)
\]
3. **Determine the Direction Ratios of Line \(AD\)**:
The direction ratios of line \(AD\) can be found by subtracting the coordinates of \(A\) from those of \(D\):
\[
\text{Direction ratios} = D - A = \left( \frac{\lambda - 1}{2} - 2, 4 - 3, \frac{5 + \mu}{2} - 5 \right)
\]
Simplifying this gives:
\[
= \left( \frac{\lambda - 1 - 4}{2}, 1, \frac{5 + \mu - 10}{2} \right) = \left( \frac{\lambda - 5}{2}, 1, \frac{\mu - 5}{2} \right)
\]
4. **Set Up the Condition for Equal Inclination**:
Since the median \(AD\) is equally inclined to the positive direction of the axes, the direction ratios must be proportional:
\[
\frac{\lambda - 5}{2} : 1 : \frac{\mu - 5}{2}
\]
This implies:
\[
\frac{\lambda - 5}{2} = 1 = \frac{\mu - 5}{2}
\]
5. **Solve the Equations**:
From the first equation:
\[
\frac{\lambda - 5}{2} = 1 \implies \lambda - 5 = 2 \implies \lambda = 7
\]
From the second equation:
\[
1 = \frac{\mu - 5}{2} \implies \mu - 5 = 2 \implies \mu = 7
\]
6. **Calculate \(2\lambda - \mu\)**:
Now substituting the values of \(\lambda\) and \(\mu\):
\[
2\lambda - \mu = 2(7) - 7 = 14 - 7 = 7
\]
### Final Answer
The value of \(2\lambda - \mu\) is **7**.
