If the direction ratios of two lines are `(1, lambda, 2) and (lambda, lambda + 1, lambda) ` and the angle between the lines is `90^(@)` then
the modulus of sum of all values of `lambda` is ________

Answer

To solve the problem, we need to find the values of \( \lambda \) such that the angle between the two lines with direction ratios \( (1, \lambda, 2) \) and \( (\lambda, \lambda + 1, \lambda) \) is \( 90^\circ \).

### Step-by-Step Solution:

1. **Understand the Condition for Perpendicular Lines:**
For two lines with direction ratios \( (a_1, b_1, c_1) \) and \( (a_2, b_2, c_2) \) to be perpendicular, the following condition must hold:
\[ a_1 \cdot a_2 + b_1 \cdot b_2 + c_1 \cdot c_2 = 0 \]

2. **Identify the Direction Ratios:**
The direction ratios of the first line are:
\[ (1, \lambda, 2) \]
The direction ratios of the second line are:
\[ (\lambda, \lambda + 1, \lambda) \]

3. **Substitute the Direction Ratios into the Perpendicularity Condition:**
Substitute the values into the condition:
\[ 1 \cdot \lambda + \lambda \cdot (\lambda + 1) + 2 \cdot \lambda = 0 \]

4. **Simplify the Equation:**
Expanding the equation gives:
\[ \lambda + \lambda^2 + \lambda + 2\lambda = 0 \]
Combine like terms:
\[ \lambda^2 + 4\lambda = 0 \]

5. **Factor the Equation:**
Factor out \( \lambda \):
\[ \lambda(\lambda + 4) = 0 \]

6. **Find the Values of \( \lambda \):**
Setting each factor to zero gives:
\[ \lambda = 0 \quad \text{or} \quad \lambda + 4 = 0 \implies \lambda = -4 \]

7. **Calculate the Sum of All Values of \( \lambda \):**
The values of \( \lambda \) are \( 0 \) and \( -4 \). Therefore, the sum is:
\[ 0 + (-4) = -4 \]

8. **Find the Modulus of the Sum:**
The modulus of the sum is:
\[ | -4 | = 4 \]

### Final Answer:
The modulus of the sum of all values of \( \lambda \) is \( 4 \).