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If q(1),q(2),q(3) ar rootsof the equatio...

If `q_(1),q_(2),q_(3)` ar rootsof the equation `x^(3)+64=0` then value of `|{:(q_(1),q_(2),q_(3)),(q_(2),q_(3),q_(1)),(q_(3),q_(1),q_(2)):}|`

A

1

B

4

C

16

D

0

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To solve the problem, we need to find the value of the determinant given the roots of the equation \( x^3 + 64 = 0 \). ### Step-by-Step Solution: 1. **Identify the roots**: The equation can be rewritten as \( x^3 = -64 \). The roots of this equation can be found by taking the cube root of \(-64\). \[ x = \sqrt[3]{-64} = -4 \] The roots of the equation \( x^3 + 64 = 0 \) are \( q_1 = -4 \), \( q_2 = -4 \), and \( q_3 = -4 \) (since it has a triple root). 2. **Set up the determinant**: We need to evaluate the determinant: \[ D = \begin{vmatrix} q_1 & q_2 & q_3 \\ q_2 & q_3 & q_1 \\ q_3 & q_1 & q_2 \end{vmatrix} \] Substituting the values of the roots: \[ D = \begin{vmatrix} -4 & -4 & -4 \\ -4 & -4 & -4 \\ -4 & -4 & -4 \end{vmatrix} \] 3. **Calculate the determinant**: Since all rows of the determinant are identical, the determinant is equal to zero: \[ D = 0 \] 4. **Conclusion**: Therefore, the value of the determinant is: \[ |D| = 0 \]
To solve the problem, we need to find the value of the determinant given the roots of the equation \( x^3 + 64 = 0 \). ### Step-by-Step Solution: 1. **Identify the roots**: The equation can be rewritten as \( x^3 = -64 \). The roots of this equation can be found by taking the cube root of \(-64\). \[ x = \sqrt[3]{-64} = -4 ...
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