Let ` alpha, beta ` be two real roots of the equation ` cot ^ 2 x - 2 lamda cot x + 3 = 0 , lamda in R ` . If ` cot ( alpha + beta ) = (1)/(2)` , then value of ` lamda ` is :

Options

A. 1
B. 2
C. ` ( 1 ) /(2) `
D. ` (3)/(2)`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Write down the given quadratic equation
The given equation is:
\[ \cot^2 x - 2\lambda \cot x + 3 = 0 \]

### Step 2: Identify the coefficients
In the standard form of a quadratic equation \( ax^2 + bx + c = 0 \), we have:
- \( a = 1 \)
- \( b = -2\lambda \)
- \( c = 3 \)

### Step 3: Use the relationships for the roots
For a quadratic equation, if \( \alpha \) and \( \beta \) are the roots, we have:
- Sum of roots:
\[ \cot \alpha + \cot \beta = -\frac{b}{a} = 2\lambda \]
- Product of roots:
\[ \cot \alpha \cdot \cot \beta = \frac{c}{a} = 3 \]

### Step 4: Use the cotangent addition formula
We know from trigonometric identities that:
\[ \cot(\alpha + \beta) = \frac{\cot \alpha \cdot \cot \beta - 1}{\cot \alpha + \cot \beta} \]
Substituting the values we have:
\[ \cot(\alpha + \beta) = \frac{3 - 1}{2\lambda} = \frac{2}{2\lambda} = \frac{1}{\lambda} \]

### Step 5: Set up the equation with the given condition
We are given that:
\[ \cot(\alpha + \beta) = \frac{1}{2} \]
Thus, we can equate:
\[ \frac{1}{\lambda} = \frac{1}{2} \]

### Step 6: Solve for \( \lambda \)
Cross-multiplying gives:
\[ 1 \cdot 2 = 1 \cdot \lambda \implies \lambda = 2 \]

### Conclusion
The value of \( \lambda \) is:
\[ \boxed{2} \]