If one root of the equation `3x^(2)-5x+lambda=0` is the reciprocal of the other, then the value of `lambda` is

Options

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

Answer

To solve the problem, we need to find the value of \( \lambda \) in the quadratic equation \( 3x^2 - 5x + \lambda = 0 \) given that one root is the reciprocal of the other.

### Step-by-Step Solution:

1. **Let the Roots be Defined**:
Let one root be \( \alpha \). Since one root is the reciprocal of the other, the second root will be \( \frac{1}{\alpha} \).

2. **Use the Relationship of Roots**:
For a quadratic equation of the form \( ax^2 + bx + c = 0 \), the sum of the roots \( \alpha + \frac{1}{\alpha} \) and the product of the roots \( \alpha \cdot \frac{1}{\alpha} \) can be expressed in terms of the coefficients:
- Sum of the roots: \( \alpha + \frac{1}{\alpha} = -\frac{b}{a} \)
- Product of the roots: \( \alpha \cdot \frac{1}{\alpha} = \frac{c}{a} \)

3. **Identify Coefficients**:
Here, \( a = 3 \), \( b = -5 \), and \( c = \lambda \).
- Therefore, the sum of the roots is:
\[ \alpha + \frac{1}{\alpha} = -\frac{-5}{3} = \frac{5}{3} \]
- The product of the roots is:
\[ \alpha \cdot \frac{1}{\alpha} = 1 = \frac{\lambda}{3} \]

4. **Set Up the Equation for Product of Roots**:
From the product of the roots, we have:
\[ 1 = \frac{\lambda}{3} \]

5. **Solve for \( \lambda \)**:
To find \( \lambda \), multiply both sides of the equation by 3:
\[ \lambda = 3 \]

### Final Answer:
The value of \( \lambda \) is \( 3 \).

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