To solve the problem, we need to find the roots of the polynomial \(27x^4 - 195x^3 + 494x^2 - 520x + 192 = 0\) under the condition that the roots are in geometric progression (G.P.).
### Step 1: Understanding the Roots in G.P.
Let the roots be \(a, ar, ar^2, ar^3\) where \(a\) is the first term and \(r\) is the common ratio of the G.P.
### Step 2: Using Vieta's Formulas
According to Vieta's formulas, for a polynomial of the form \(ax^4 + bx^3 + cx^2 + dx + e = 0\), the relationships between the coefficients and the roots are as follows:
- The sum of the roots: \(a + ar + ar^2 + ar^3 = -\frac{b}{a}\)
- The sum of the products of the roots taken two at a time: \(a \cdot ar + a \cdot ar^2 + a \cdot ar^3 + ar \cdot ar^2 + ar \cdot ar^3 + ar^2 \cdot ar^3 = \frac{c}{a}\)
- The product of the roots: \(a \cdot ar \cdot ar^2 \cdot ar^3 = \frac{e}{a}\)
### Step 3: Calculate the Product of the Roots
The product of the roots is given by:
\[
a \cdot ar \cdot ar^2 \cdot ar^3 = a^4 r^6
\]
According to Vieta's, this is equal to:
\[
\frac{192}{27} = \frac{64}{9}
\]
Thus, we have:
\[
a^4 r^6 = \frac{64}{9}
\]
### Step 4: Calculate the Sum of the Roots
The sum of the roots is:
\[
a(1 + r + r^2 + r^3) = a \cdot \frac{r^4 - 1}{r - 1} = -\frac{-195}{27} = \frac{195}{27} = \frac{65}{9}
\]
### Step 5: Calculate the Sum of the Products of the Roots
The sum of the products of the roots taken two at a time is:
\[
a^2(r + r^2 + r^3 + r^2 + r^3 + r^4) = a^2 \cdot \frac{r^6 - 1}{r - 1} = \frac{494}{27}
\]
### Step 6: Solving for Roots
Now we need to find values of \(a\) and \(r\) that satisfy these equations.
From the product equation:
\[
a^4 r^6 = \frac{64}{9}
\]
Assuming \(a = \frac{2}{3}\) and \(r = 2\), we can check:
\[
\left(\frac{2}{3}\right)^4 \cdot 2^6 = \frac{16}{81} \cdot 64 = \frac{1024}{81} \neq \frac{64}{9}
\]
We can try different values for \(a\) and \(r\) until we find a combination that satisfies all equations.
### Step 7: Testing Options
Given the options, we can test the combinations:
1. \(2, 3, \frac{4}{3}, \frac{8}{9}\)
2. \(1, 2, 3, 4\)
3. \(4, 2, \frac{8}{9}, \frac{4}{3}\)
After testing, we find that:
- The combination \(2, \frac{4}{3}, \frac{8}{9}, 3\) satisfies all conditions.
### Conclusion
Thus, the roots of the equation \(27x^4 - 195x^3 + 494x^2 - 520x + 192 = 0\) that are in G.P. are:
\[
\frac{8}{9}, \frac{4}{3}, 2, 3
\]
