Out of the two roots of `x^(2) + (1 - 2lambda )x + (lambda^(2)-lambda-2)=0` one root is greater than 3 and the other root is less than 3, then the limits of `lambda` are

Options

A. `lambda lt 2`
B. `2 lt lambda lt 5`
C. ` lambda gt 5`
D. `lambda= (5)/(2)`

Answer

To solve the problem, we need to find the limits of \( \lambda \) such that one root of the quadratic equation \( x^2 + (1 - 2\lambda)x + (\lambda^2 - \lambda - 2) = 0 \) is greater than 3 and the other root is less than 3.

### Step-by-step Solution:

1. **Identify the quadratic equation**:
The given quadratic equation is:
\[ f(x) = x^2 + (1 - 2\lambda)x + (\lambda^2 - \lambda - 2) \]

2. **Determine the condition for the roots**:
Since one root is greater than 3 and the other is less than 3, it implies that the value of the function at \( x = 3 \) must be negative:
\[ f(3) < 0 \]

3. **Calculate \( f(3) \)**:
Substitute \( x = 3 \) into the function:
\[ f(3) = 3^2 + (1 - 2\lambda) \cdot 3 + (\lambda^2 - \lambda - 2) \]
Simplifying this gives:
\[ f(3) = 9 + 3 - 6\lambda + \lambda^2 - \lambda - 2 \]
\[ = \lambda^2 - 7\lambda + 10 \]

4. **Set up the inequality**:
We need to solve the inequality:
\[ \lambda^2 - 7\lambda + 10 < 0 \]

5. **Factor the quadratic**:
To factor \( \lambda^2 - 7\lambda + 10 \):
\[ \lambda^2 - 7\lambda + 10 = (\lambda - 2)(\lambda - 5) \]

6. **Determine the intervals**:
The inequality \( (\lambda - 2)(\lambda - 5) < 0 \) is satisfied between the roots:
- The roots are \( \lambda = 2 \) and \( \lambda = 5 \).
- The intervals to test are \( (-\infty, 2) \), \( (2, 5) \), and \( (5, \infty) \).

7. **Test the intervals**:
- For \( \lambda < 2 \) (e.g., \( \lambda = 0 \)): \( (0 - 2)(0 - 5) = 10 > 0 \)
- For \( 2 < \lambda < 5 \) (e.g., \( \lambda = 3 \)): \( (3 - 2)(3 - 5) = -2 < 0 \)
- For \( \lambda > 5 \) (e.g., \( \lambda = 6 \)): \( (6 - 2)(6 - 5) = 4 > 0 \)

8. **Conclusion**:
The inequality \( (\lambda - 2)(\lambda - 5) < 0 \) holds true for:
\[ 2 < \lambda < 5 \]

### Final Answer:
The limits of \( \lambda \) are:
\[ \boxed{2 < \lambda < 5} \]