If A, B, C, D are the sum of the square of the roots of `2x^(2) + x- 3=0, x^(2)-x+2= 0, 3x^(2) - 2x +1=0, x^(2)- x+1=0` then the ascending order of A, B, C, D is

Options

A. `B lt D lt C lt A`
B. `B lt C lt A lt D`
C. `C lt B lt A lt D`
D. `Alt D lt C lt B`

Answer

To solve the problem, we need to find the sum of the squares of the roots for each of the given quadratic equations and then arrange those sums in ascending order.

### Step-by-Step Solution:

1. **Identify the Quadratic Equations**:
- The equations given are:
1. \(2x^2 + x - 3 = 0\)
2. \(x^2 - x + 2 = 0\)
3. \(3x^2 - 2x + 1 = 0\)
4. \(x^2 - x + 1 = 0\)

2. **Use the Formula for the Sum of the Squares of the Roots**:
The sum of the squares of the roots \(S\) of a quadratic equation \(Ax^2 + Bx + C = 0\) can be calculated using the formula:
\[ S = \frac{B^2 - 2AC}{A^2} \]

3. **Calculate for Each Equation**:

- **For the first equation \(2x^2 + x - 3 = 0\)**:
- Here, \(A = 2\), \(B = 1\), \(C = -3\).
\[ S_A = \frac{1^2 - 2 \cdot 2 \cdot (-3)}{2^2} = \frac{1 + 12}{4} = \frac{13}{4} \]

- **For the second equation \(x^2 - x + 2 = 0\)**:
- Here, \(A = 1\), \(B = -1\), \(C = 2\).
\[ S_B = \frac{(-1)^2 - 2 \cdot 1 \cdot 2}{1^2} = \frac{1 - 4}{1} = -3 \]

- **For the third equation \(3x^2 - 2x + 1 = 0\)**:
- Here, \(A = 3\), \(B = -2\), \(C = 1\).
\[ S_C = \frac{(-2)^2 - 2 \cdot 3 \cdot 1}{3^2} = \frac{4 - 6}{9} = \frac{-2}{9} \]

- **For the fourth equation \(x^2 - x + 1 = 0\)**:
- Here, \(A = 1\), \(B = -1\), \(C = 1\).
\[ S_D = \frac{(-1)^2 - 2 \cdot 1 \cdot 1}{1^2} = \frac{1 - 2}{1} = -1 \]

4. **Summarize the Results**:
- \(A = \frac{13}{4}\)
- \(B = -3\)
- \(C = -\frac{2}{9}\)
- \(D = -1\)

5. **Arrange in Ascending Order**:
- Convert all values to a common format for comparison:
- \(B = -3\)
- \(D = -1\)
- \(C = -\frac{2}{9} \approx -0.222\)
- \(A = \frac{13}{4} = 3.25\)

- The ascending order is:
\[ B < D < C < A \quad \text{or} \quad -3 < -1 < -\frac{2}{9} < \frac{13}{4} \]

### Final Answer:
The ascending order of \(A, B, C, D\) is:
\[ B < D < C < A \]