The values of `lamda` for which one root of the equation `x^2+(1-2lamda)x+(lamda^2-lamda-2)=0` is greater than 3 and the other smaller than 2 are given by

Options

A. `2ltlamdalt5`
B. `1ltlamdalt4`
C. `1ltlamdalt5`
D. `2ltlamdalt4`

Answer

To solve the problem, we need to find the values of \( \lambda \) for which 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 2.

### Step 1: Identify the roots of the quadratic equation
The roots of the quadratic equation can be found using the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

where \( a = 1 \), \( b = 1 - 2\lambda \), and \( c = \lambda^2 - \lambda - 2 \).

### Step 2: Set up the conditions for the roots
Let the roots be \( r_1 \) and \( r_2 \). We want:

1. \( r_1 > 3 \)
2. \( r_2 < 2 \)

### Step 3: Calculate \( f(2) \) and \( f(3) \)
To find the conditions on \( \lambda \), we will evaluate the quadratic at \( x = 2 \) and \( x = 3 \).

#### For \( x = 2 \):
\[ f(2) = 2^2 + (1 - 2\lambda)(2) + (\lambda^2 - \lambda - 2) \]
\[ = 4 + (2 - 4\lambda) + (\lambda^2 - \lambda - 2) \]
\[ = \lambda^2 - 5\lambda + 4 < 0 \]

#### For \( x = 3 \):
\[ f(3) = 3^2 + (1 - 2\lambda)(3) + (\lambda^2 - \lambda - 2) \]
\[ = 9 + (3 - 6\lambda) + (\lambda^2 - \lambda - 2) \]
\[ = \lambda^2 - 7\lambda + 10 < 0 \]

### Step 4: Solve the inequalities
1. **For \( f(2) < 0 \)**:
\[ \lambda^2 - 5\lambda + 4 < 0 \]
Factoring gives:
\[ (\lambda - 4)(\lambda - 1) < 0 \]
The solution is \( 1 < \lambda < 4 \).

2. **For \( f(3) < 0 \)**:
\[ \lambda^2 - 7\lambda + 10 < 0 \]
Factoring gives:
\[ (\lambda - 5)(\lambda - 2) < 0 \]
The solution is \( 2 < \lambda < 5 \).

### Step 5: Find the intersection of the intervals
We have two intervals:
- From \( f(2) < 0 \): \( 1 < \lambda < 4 \)
- From \( f(3) < 0 \): \( 2 < \lambda < 5 \)

The intersection of these intervals is:
\[ 2 < \lambda < 4 \]

### Conclusion
Thus, the values of \( \lambda \) for which one root is greater than 3 and the other is less than 2 are:

\[ \lambda \in (2, 4) \]