Statement -1 : Let a,b,c are the sides of a `DeltaABC` such that `|{:(a,a^2b-1,a^2+c),(b,b^3-1,b^2+c),(c,c^2b-1,c^2+c):}|=0` then triangle is isosceles.
Statement -2 : All the element of any two rows are equal then the value of the determinant is equal to zero.

Options

A. Statement -1 is True, Statement -2 is True, Statement -2 is a correct explanation for statement-1
B. Statement -1 is true, statement-2 is true , statement-2 is NOT a correct explanation for statement-1
C. Statement-1 is true, statement -2 is False
D. Statement -1 is False, Statement -2 is true

Answer

To solve the problem, we need to analyze the given determinant and prove the statements regarding the triangle being isosceles and the properties of determinants.

### Step-by-Step Solution:

1. **Write Down the Determinant**:
We have the determinant:
\[ D = \begin{vmatrix} a & a^2b - 1 & a^2 + c \\ b & b^3 - 1 & b^2 + c \\ c & c^2b - 1 & c^2 + c \end{vmatrix} \]

2. **Assume \( a = b \)**:
To check if the triangle is isosceles, we assume \( a = b \). This means we can replace \( b \) with \( a \) in the determinant:
\[ D = \begin{vmatrix} a & a^2a - 1 & a^2 + c \\ a & a^3 - 1 & a^2 + c \\ c & c^2a - 1 & c^2 + c \end{vmatrix} \]

3. **Simplify the Determinant**:
The first two rows of the determinant now are:
- Row 1: \( (a, a^3 - 1, a^2 + c) \)
- Row 2: \( (a, a^3 - 1, a^2 + c) \)

Since the first two rows are identical, we can apply the property of determinants that states if two rows are the same, the determinant is zero:
\[ D = 0 \]

4. **Conclusion for Statement 1**:
Since we have shown that \( D = 0 \) when \( a = b \), this implies that the triangle is isosceles. Therefore, Statement 1 is true.

5. **Verify Statement 2**:
Statement 2 states that if all elements of any two rows are equal, then the value of the determinant is equal to zero. This is indeed a property of determinants. Thus, Statement 2 is also true.

### Final Result:
Both statements are true:
- Statement 1: True (the triangle is isosceles).
- Statement 2: True (property of determinants).