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 smaller than 2.

### Step 1: Identify the quadratic function
The given quadratic function can be expressed as:

\[ f(x) = x^2 + (1 - 2\lambda)x + (\lambda^2 - \lambda - 2) \]

### Step 2: Determine conditions for roots
Since the parabola opens upwards (as the coefficient of \(x^2\) is positive), we need to ensure that:

- \(f(2) < 0\) (indicating one root is less than 2)
- \(f(3) < 0\) (indicating the other root is greater than 3)

### Step 3: Calculate \(f(2)\)
Substituting \(x = 2\) into the function:

\[ f(2) = 2^2 + (1 - 2\lambda) \cdot 2 + (\lambda^2 - \lambda - 2) \]

Calculating this gives:

\[ f(2) = 4 + (2 - 4\lambda) + (\lambda^2 - \lambda - 2) \]
\[ = \lambda^2 - 5\lambda + 6 < 0 \]

### Step 4: Factor the quadratic inequality
We can factor the quadratic:

\[ \lambda^2 - 5\lambda + 6 = (\lambda - 2)(\lambda - 3) < 0 \]

### Step 5: Solve the inequality
The roots of the quadratic are \(\lambda = 2\) and \(\lambda = 3\). The inequality \((\lambda - 2)(\lambda - 3) < 0\) holds true for:

\[ 2 < \lambda < 3 \]

### Step 6: Calculate \(f(3)\)
Now substituting \(x = 3\) into the function:

\[ f(3) = 3^2 + (1 - 2\lambda) \cdot 3 + (\lambda^2 - \lambda - 2) \]

Calculating this gives:

\[ f(3) = 9 + (3 - 6\lambda) + (\lambda^2 - \lambda - 2) \]
\[ = \lambda^2 - 7\lambda + 10 < 0 \]

### Step 7: Factor the second quadratic inequality
We can factor the quadratic:

\[ \lambda^2 - 7\lambda + 10 = (\lambda - 2)(\lambda - 5) < 0 \]

### Step 8: Solve the second inequality
The roots of this quadratic are \(\lambda = 2\) and \(\lambda = 5\). The inequality \((\lambda - 2)(\lambda - 5) < 0\) holds true for:

\[ 2 < \lambda < 5 \]

### Step 9: Find the intersection of both conditions
From the two inequalities we have:

1. \(2 < \lambda < 3\)
2. \(2 < \lambda < 5\)

The intersection of these two conditions is:

\[ 2 < \lambda < 3 \]

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

\[ \lambda \in (2, 3) \]