The direction ratios of a line are (-2, 3, 6). If the line makes an acute angle with positive direction of x-axis
then the modulus of integral value of sum of all direction cosines, is _________.

Answer

To solve the problem, we need to find the modulus of the integral value of the sum of all direction cosines given the direction ratios of the line are (-2, 3, 6) and that the line makes an acute angle with the positive direction of the x-axis.

### Step-by-step Solution:

1. **Identify Direction Ratios**:
The direction ratios of the line are given as \( a = -2 \), \( b = 3 \), and \( c = 6 \).

2. **Calculate the Magnitude**:
First, we need to find the magnitude of the direction ratios:
\[ \sqrt{a^2 + b^2 + c^2} = \sqrt{(-2)^2 + 3^2 + 6^2} = \sqrt{4 + 9 + 36} = \sqrt{49} = 7. \]

3. **Calculate Direction Cosines**:
The direction cosines \( l, m, n \) are given by:
\[ l = \frac{a}{\sqrt{a^2 + b^2 + c^2}}, \quad m = \frac{b}{\sqrt{a^2 + b^2 + c^2}}, \quad n = \frac{c}{\sqrt{a^2 + b^2 + c^2}}. \]
Substituting the values:
\[ l = \frac{-2}{7}, \quad m = \frac{3}{7}, \quad n = \frac{6}{7}. \]

4. **Sum of Direction Cosines**:
Now, we calculate the sum of the direction cosines:
\[ l + m + n = \frac{-2}{7} + \frac{3}{7} + \frac{6}{7}. \]
Finding a common denominator (which is 7):
\[ l + m + n = \frac{-2 + 3 + 6}{7} = \frac{7}{7} = 1. \]

5. **Find the Modulus**:
The modulus of the sum of the direction cosines is:
\[ |l + m + n| = |1| = 1. \]

6. **Final Answer**:
Therefore, the modulus of the integral value of the sum of all direction cosines is:
\[ \boxed{1}. \]