Which of the following are the roots of `3x^(2)+2x-1=0`?
(i) `-1" "(ii)" "(1)/(3)" "(iii)" "-(1)/(2)`

Answer

To determine which of the given options are the roots of the quadratic equation \(3x^2 + 2x - 1 = 0\), we will substitute each option into the equation and check if it satisfies the equation.

### Step-by-Step Solution:

1. **Substituting \(x = -1\)**:
\[ 3(-1)^2 + 2(-1) - 1 = 3(1) - 2 - 1 = 3 - 2 - 1 = 0 \]
Since the left-hand side equals 0, \(x = -1\) is a root.

2. **Substituting \(x = \frac{1}{3}\)**:
\[ 3\left(\frac{1}{3}\right)^2 + 2\left(\frac{1}{3}\right) - 1 = 3\left(\frac{1}{9}\right) + \frac{2}{3} - 1 = \frac{1}{3} + \frac{2}{3} - 1 = 1 - 1 = 0 \]
Since the left-hand side equals 0, \(x = \frac{1}{3}\) is also a root.

3. **Substituting \(x = -\frac{1}{2}\)**:
\[ 3\left(-\frac{1}{2}\right)^2 + 2\left(-\frac{1}{2}\right) - 1 = 3\left(\frac{1}{4}\right) - 1 - 1 = \frac{3}{4} - 1 - 1 = \frac{3}{4} - \frac{4}{4} - \frac{4}{4} = \frac{3 - 4 - 4}{4} = \frac{-5}{4} \neq 0 \]
Since the left-hand side does not equal 0, \(x = -\frac{1}{2}\) is not a root.

### Conclusion:
The roots of the equation \(3x^2 + 2x - 1 = 0\) are \(x = -1\) and \(x = \frac{1}{3}\). Therefore, the correct options are (i) and (ii).