Let `A(2hat(i)+3hat(j)+5hat(k)), B(-hat(i)+3hat(j)+2hat(k)) and C(lambdahat(i)+5hat(j)+muhat(k))` are vertices of a triangle and its median through A is equally inclined to the positive directions of the axes, the value of `2lambda-mu` is equal to

Answer

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 directions of the axes.

### Step-by-step Solution:

1. **Identify the Coordinates of Points A, B, and C:**
- \(A = (2, 3, 5)\)
- \(B = (-1, 3, 2)\)
- \(C = (\lambda, 5, \mu)\)

2. **Find the Midpoint D of Segment BC:**
The midpoint \(D\) of segment \(BC\) can be calculated using the midpoint formula:
\[ D = \left(\frac{x_B + x_C}{2}, \frac{y_B + y_C}{2}, \frac{z_B + z_C}{2}\right) \]
Substituting the coordinates of \(B\) and \(C\):
\[ D = \left(\frac{-1 + \lambda}{2}, \frac{3 + 5}{2}, \frac{2 + \mu}{2}\right) = \left(\frac{\lambda - 1}{2}, 4, \frac{\mu + 2}{2}\right) \]

3. **Determine the Direction Ratios of Median AD:**
The direction ratios of the median \(AD\) can be found by subtracting the coordinates of \(A\) from \(D\):
\[ \text{Direction Ratios of } AD = D - A = \left(\frac{\lambda - 1}{2} - 2, 4 - 3, \frac{\mu + 2}{2} - 5\right) \]
Simplifying this gives:
\[ = \left(\frac{\lambda - 5}{2}, 1, \frac{\mu - 8}{2}\right) \]

4. **Set Up the Condition for Equal Inclination:**
Since the median \(AD\) is equally inclined to the positive directions of the axes, the direction ratios must be equal:
\[ \frac{\lambda - 5}{2} = 1 \quad \text{and} \quad 1 = \frac{\mu - 8}{2} \]

5. **Solve for \(\lambda\) and \(\mu\):**
From the first equation:
\[ \frac{\lambda - 5}{2} = 1 \implies \lambda - 5 = 2 \implies \lambda = 7 \]
From the second equation:
\[ 1 = \frac{\mu - 8}{2} \implies \mu - 8 = 2 \implies \mu = 10 \]

6. **Calculate \(2\lambda - \mu\):**
Now substitute the values of \(\lambda\) and \(\mu\):
\[ 2\lambda - \mu = 2(7) - 10 = 14 - 10 = 4 \]

### Final Answer:
The value of \(2\lambda - \mu\) is \(4\).