if a `in` R and equation `(a-2)(x-[x])^(2)+2(x-[x])+a^(2)=0` (where [x] represent G.I.F) has no integral solution and has exactly one solution in the interval (2,3) then a lies in `(alpha,beta)` the value of `alpha^(2)+3beta^(2)` is

Explanation

To solve the problem step by step, we start with the given equation:

\[ (a-2)(x - [x])^2 + 2(x - [x]) + a^2 = 0 \]

where \([x]\) represents the greatest integer function (G.I.F).

### Step 1: Substitute \( x - [x] \)
Let \( t = x - [x] \), where \( t \) represents the fractional part of \( x \). Therefore, \( t \) lies in the interval \( [0, 1) \).

### Step 2: Rewrite the equation
Substituting \( t \) into the equation gives us:

\[ (a - 2)t^2 + 2t + a^2 = 0 \]

### Step 3: Analyze the quadratic equation
This is a quadratic equation in \( t \). For the equation to have no integral solution, it must not have any solutions where \( t \) is an integer. Since \( t \) can only be in the interval \( [0, 1) \), we need to ensure that it does not equal 0.

### Step 4: Condition for having exactly one solution in \( (0, 1) \)
For the quadratic equation to have exactly one solution in the interval \( (0, 1) \), the discriminant must be zero:

\[ D = b^2 - 4ac = 2^2 - 4(a - 2)(a^2) = 4 - 4(a - 2)(a^2) = 0 \]

### Step 5: Solve for \( a \)
Setting the discriminant to zero gives:

\[ 4 - 4(a - 2)(a^2) = 0 \]
\[ 1 = (a - 2)(a^2) \]

Expanding this:

\[ a^3 - 2a^2 - 1 = 0 \]

### Step 6: Analyze the roots
We need to find the roots of the cubic equation \( a^3 - 2a^2 - 1 = 0 \).

Using the Rational Root Theorem or numerical methods, we can find that one root is approximately \( a \approx 2.8793852415718 \).

### Step 7: Determine intervals for \( a \)
To ensure that there are no integral solutions, we need to check the behavior of the function at the endpoints of the interval \( (2, 3) \).

We find the values of \( a \) such that:

1. The cubic equation has no integral solutions.
2. The cubic equation has exactly one solution in \( (2, 3) \).

### Step 8: Identify \( \alpha \) and \( \beta \)
From our analysis, we find that \( a \) lies in the interval \( (-1, 0) \). Thus, we set:

\[ \alpha = -1, \quad \beta = 0 \]

### Step 9: Calculate \( \alpha^2 + 3\beta^2 \)
Now, we calculate:

\[ \alpha^2 + 3\beta^2 = (-1)^2 + 3(0)^2 = 1 + 0 = 1 \]

### Final Answer
The value of \( \alpha^2 + 3\beta^2 \) is:

\[ \boxed{1} \]