One of the roots of a quadratic equation with real coefficients is `(1)/((2-3i))`. Which of the following implications is/are true?
1. The second root of the equation will be `(1)/((3-2i))`.
2. The equation has no real root.
3. The equation is `13x^(2)-4x+1=0.`
Which of the above is/are correct ?

Options

A. 1 and 2 only
B. 3 only
C. 2 and 3 only
D. 1,2 and 3

Explanation

To solve the problem, we need to analyze the implications of having one root of a quadratic equation as \( \frac{1}{2 - 3i} \).

### Step-by-step Solution:

1. **Identify the Given Root:**
The given root is \( r_1 = \frac{1}{2 - 3i} \).

2. **Find the Conjugate Root:**
Since the coefficients of the quadratic equation are real, the complex roots must occur in conjugate pairs. Therefore, the conjugate of \( r_1 \) is:
\[ r_2 = \frac{1}{2 + 3i} \]

3. **Check the First Implication:**
The first implication states that the second root of the equation will be \( \frac{1}{3 - 2i} \). This is incorrect because the second root must be the conjugate of the first root, which we found to be \( \frac{1}{2 + 3i} \).

4. **Check the Second Implication:**
The second implication states that the equation has no real roots. Since both roots are complex (they are conjugates), this implication is true.

5. **Check the Third Implication:**
The third implication states that the equation is \( 13x^2 - 4x + 1 = 0 \). To verify this, we need to check the sum and product of the roots:
- **Sum of the roots**:
\[ r_1 + r_2 = \frac{1}{2 - 3i} + \frac{1}{2 + 3i} \]
To compute this, we find a common denominator:
\[ = \frac{(2 + 3i) + (2 - 3i)}{(2 - 3i)(2 + 3i)} = \frac{4}{4 + 9} = \frac{4}{13} \]
According to Vieta's formulas, the sum of the roots \( -\frac{b}{a} \) should equal \( \frac{4}{13} \).

- **Product of the roots**:
\[ r_1 \cdot r_2 = \left(\frac{1}{2 - 3i}\right) \left(\frac{1}{2 + 3i}\right) = \frac{1}{(2 - 3i)(2 + 3i)} = \frac{1}{4 + 9} = \frac{1}{13} \]
According to Vieta's formulas, the product of the roots \( \frac{c}{a} \) should equal \( \frac{1}{13} \).

Since both the sum and product of the roots match the coefficients of the quadratic equation \( 13x^2 - 4x + 1 = 0 \), this implication is also true.

### Conclusion:
- The first implication is **false**.
- The second implication is **true**.
- The third implication is **true**.

Thus, the correct implications are the second and third.