Correct Answer - Option 3 :

\(\frac{7}{2}\)
**Concept:**

Consider a quadratic equation: ax^{2} + bx + c = 0.

Let, α and β are the roots.

- Sum of roots = α + β = -b/a
- Product of the roots = αβ = c/a

\(\rm (a+b)^2=(a-b)^2+4ab\)

**Calculation:**

Given equation: \(\rm 2x^2+5x-3=0\)

α and β are the roots of the given equation.

∴ Sum of roots = α + β = \(-\frac{5}{2}\) and

Product of the roots = αβ = \(\frac{-3}{2}\)

Now, we know, \(\rm (a+b)^2=(a-b)^2+4ab\)

∴ (α + β)^{2} = (α - β)^{2} + 4αβ

⇒ \(\rm (-\frac52)^2=(\alpha -\beta)^2+4(-\frac32)\)

\(\Rightarrow \rm (\alpha -\beta)^2=\frac{25}{4}+6\)

\(=\frac{49}{4}\)

\(\Rightarrow \rm |(\alpha -\beta)|=\frac{7}{2}\)

Hence, option (3) is correct.