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 will follow these steps:

### Step 1: Understand Direction Ratios and Direction Cosines
The direction ratios of the line are given as (-2, 3, 6). The direction cosines (L, M, N) are calculated using the formula:
\[ L = \frac{a}{\sqrt{a^2 + b^2 + c^2}}, \quad M = \frac{b}{\sqrt{a^2 + b^2 + c^2}}, \quad N = \frac{c}{\sqrt{a^2 + b^2 + c^2}} \]
where \( a, b, c \) are the direction ratios.

### Step 2: Calculate the Magnitude of Direction Ratios
First, we need to calculate the magnitude of the direction ratios:
\[ \sqrt{(-2)^2 + 3^2 + 6^2} = \sqrt{4 + 9 + 36} = \sqrt{49} = 7 \]

### Step 3: Calculate Direction Cosines
Now we can calculate the direction cosines:
- For L:
\[ L = \frac{-2}{7} \]
- For M:
\[ M = \frac{3}{7} \]
- For N:
\[ N = \frac{6}{7} \]

### Step 4: Sum of Direction Cosines
Next, we find the sum of the direction cosines:
\[ L + M + N = \frac{-2}{7} + \frac{3}{7} + \frac{6}{7} \]

### Step 5: Simplify the Sum
Now, simplifying the sum:
\[ L + M + N = \frac{-2 + 3 + 6}{7} = \frac{7}{7} = 1 \]

### Step 6: Calculate the Modulus
Finally, we need the modulus of the sum:
\[ |L + M + N| = |1| = 1 \]

### Final Answer
The modulus of the integral value of the sum of all direction cosines is:
\[ \boxed{1} \]

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