If ratio of the roots of the quadratic equation `3m^2x^2+m(m-4)x+2=0` is `lamda` such that `lamda+1/lamda=1` then least value of `m` is (A) `-2-2sqrt3` (B) `-2+2sqrt3` (C) `4+3sqrt2` (D) `4-3sqrt2`

Options

A. `2 - sqrt(3)`
B. `4 - 3sqrt(2)`
C. `-2 + sqrt(2)`
D. `4 - 2sqrt(3)`

Answer

To solve the problem, we need to find the least value of \( m \) for the quadratic equation \( 3m^2x^2 + m(m-4)x + 2 = 0 \) given that the ratio of the roots \( \lambda \) satisfies \( \lambda + \frac{1}{\lambda} = 1 \).

### Step-by-Step Solution:

1. **Understanding the Roots**:
Let the roots of the quadratic equation be \( \alpha \) and \( \beta \). The ratio of the roots is given by \( \lambda = \frac{\alpha}{\beta} \).

2. **Using the Given Condition**:
The condition \( \lambda + \frac{1}{\lambda} = 1 \) can be rewritten as:
\[ \frac{\alpha}{\beta} + \frac{\beta}{\alpha} = 1 \]
Multiplying through by \( \alpha \beta \) gives:
\[ \alpha^2 + \beta^2 = \alpha \beta \]

3. **Sum and Product of Roots**:
From Vieta's formulas, we know:
- Sum of the roots \( \alpha + \beta = -\frac{b}{a} = -\frac{m(m-4)}{3m^2} \)
- Product of the roots \( \alpha \beta = \frac{c}{a} = \frac{2}{3m^2} \)

4. **Expressing \( \alpha^2 + \beta^2 \)**:
We can express \( \alpha^2 + \beta^2 \) in terms of the sum and product:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]
Substituting the values:
\[ \alpha^2 + \beta^2 = \left(-\frac{m(m-4)}{3m^2}\right)^2 - 2\left(\frac{2}{3m^2}\right) \]

5. **Setting Up the Equation**:
Now we set up the equation:
\[ \left(-\frac{m(m-4)}{3m^2}\right)^2 - 2\left(\frac{2}{3m^2}\right) = \frac{2}{3m^2} \]
Simplifying gives:
\[ \frac{m^2(m-4)^2}{9m^4} - \frac{4}{3m^2} = \frac{2}{3m^2} \]
Multiplying through by \( 9m^4 \) to eliminate the denominators:
\[ m^2(m-4)^2 - 12m^2 = 6m^2 \]

6. **Rearranging the Equation**:
This simplifies to:
\[ m^2(m-4)^2 - 18m^2 = 0 \]
Factoring out \( m^2 \):
\[ m^2((m-4)^2 - 18) = 0 \]
Thus, we have:
\[ (m-4)^2 = 18 \]

7. **Solving for \( m \)**:
Taking the square root:
\[ m - 4 = \pm \sqrt{18} = \pm 3\sqrt{2} \]
Therefore:
\[ m = 4 + 3\sqrt{2} \quad \text{or} \quad m = 4 - 3\sqrt{2} \]

8. **Finding the Least Value**:
The least value of \( m \) is:
\[ m = 4 - 3\sqrt{2} \]

### Final Answer:
The least value of \( m \) is \( \boxed{4 - 3\sqrt{2}} \).