If `A=|(1,1,1),(a,b,c),(a^3,b^3,c^3)|, B=|(1,1,1),(a^2,b^2,c^2),(a^3,b^3,c^3)|, C=|(a,b,c),(a^2,b^2,c^2),(a^3,b^3,c^3)|` , then which relation is correct :

Options

A. A=B
B. A=C
C. B=C
D. None of these

Answer

To solve the problem, we need to evaluate the determinants \( A \), \( B \), and \( C \) given as:

\[ A = \begin{vmatrix} 1 & 1 & 1 \\ a & b & c \\ a^3 & b^3 & c^3 \end{vmatrix}, \quad B = \begin{vmatrix} 1 & 1 & 1 \\ a^2 & b^2 & c^2 \\ a^3 & b^3 & c^3 \end{vmatrix}, \quad C = \begin{vmatrix} a & b & c \\ a^2 & b^2 & c^2 \\ a^3 & b^3 & c^3 \end{vmatrix} \]

### Step 1: Calculate Determinant \( A \)

To calculate \( A \), we can use the properties of determinants. We can expand \( A \) using the first row:

\[ A = 1 \cdot \begin{vmatrix} b & c \\ b^3 & c^3 \end{vmatrix} - 1 \cdot \begin{vmatrix} a & c \\ a^3 & c^3 \end{vmatrix} + 1 \cdot \begin{vmatrix} a & b \\ a^3 & b^3 \end{vmatrix} \]

Calculating the 2x2 determinants:

\[ \begin{vmatrix} b & c \\ b^3 & c^3 \end{vmatrix} = bc^3 - cb^3 = b(c^3 - b^2c) \]

\[ \begin{vmatrix} a & c \\ a^3 & c^3 \end{vmatrix} = ac^3 - ca^3 = a(c^3 - a^2c) \]

\[ \begin{vmatrix} a & b \\ a^3 & b^3 \end{vmatrix} = ab^3 - ba^3 = a(b^3 - a^2b) \]

Thus, we can express \( A \):

\[ A = b(c^3 - b^2c) - a(c^3 - a^2c) + a(b^3 - a^2b) \]

### Step 2: Calculate Determinant \( B \)

Next, we calculate \( B \):

\[ B = \begin{vmatrix} 1 & 1 & 1 \\ a^2 & b^2 & c^2 \\ a^3 & b^3 & c^3 \end{vmatrix} \]

Using the same expansion method:

\[ B = 1 \cdot \begin{vmatrix} b^2 & c^2 \\ b^3 & c^3 \end{vmatrix} - 1 \cdot \begin{vmatrix} a^2 & c^2 \\ a^3 & c^3 \end{vmatrix} + 1 \cdot \begin{vmatrix} a^2 & b^2 \\ a^3 & b^3 \end{vmatrix} \]

Calculating the 2x2 determinants:

\[ \begin{vmatrix} b^2 & c^2 \\ b^3 & c^3 \end{vmatrix} = b^2c^3 - c^2b^3 = b^2(c^3 - b c^2) \]

\[ \begin{vmatrix} a^2 & c^2 \\ a^3 & c^3 \end{vmatrix} = a^2c^3 - c^2a^3 = a^2(c^3 - a c^2) \]

\[ \begin{vmatrix} a^2 & b^2 \\ a^3 & b^3 \end{vmatrix} = a^2b^3 - b^2a^3 = a^2(b^3 - a b^2) \]

Thus, we can express \( B \):

\[ B = b^2(c^3 - b c^2) - a^2(c^3 - a c^2) + a^2(b^3 - a b^2) \]

### Step 3: Calculate Determinant \( C \)

Finally, we calculate \( C \):

\[ C = \begin{vmatrix} a & b & c \\ a^2 & b^2 & c^2 \\ a^3 & b^3 & c^3 \end{vmatrix} \]

Using the same expansion method:

\[ C = a \begin{vmatrix} b^2 & c^2 \\ b^3 & c^3 \end{vmatrix} - b \begin{vmatrix} a^2 & c^2 \\ a^3 & c^3 \end{vmatrix} + c \begin{vmatrix} a^2 & b^2 \\ a^3 & b^3 \end{vmatrix} \]

The calculations for the 2x2 determinants are the same as before. Thus, we can express \( C \):

\[ C = a \cdot b^2(c^3 - b c^2) - b \cdot a^2(c^3 - a c^2) + c \cdot a^2(b^3 - a b^2) \]

### Conclusion

After calculating the determinants \( A \), \( B \), and \( C \), we can analyze their relationships. The relationships can be summarized as follows:

- \( A \) and \( C \) have a direct relationship.
- \( B \) does not have a direct relationship with \( A \) or \( C \).

Thus, the correct relation is that \( C \) is proportional to \( A \), but there is no direct relationship between \( A \) and \( B \).