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If q(1), (2), q(3) are roots of the eq...

If `q_(1), _(2), q_(3)` are roots of the equation `x^(3)+64=0`, then the value of `|(q_(1),q_(2),q_(3)),(q_(2),q_(3), q_(1)),(q_(3),q_(1),q_(2))|` is :-

A

1

B

4

C

16

D

0

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The correct Answer is:
To solve the problem, we need to find the value of the determinant: \[ D = \begin{vmatrix} q_1 & q_2 & q_3 \\ q_2 & q_3 & q_1 \\ q_3 & q_1 & q_2 \end{vmatrix} \] where \( q_1, q_2, q_3 \) are the roots of the equation \( x^3 + 64 = 0 \). ### Step 1: Find the roots of the equation The equation can be rewritten as: \[ x^3 = -64 \] Taking the cube root of both sides, we find: \[ x = \sqrt[3]{-64} = -4 \] The roots of the equation are \( q_1 = -4 \), \( q_2 = -4 \), and \( q_3 = -4 \). ### Step 2: Substitute the roots into the determinant Now substitute \( q_1, q_2, q_3 \) into the determinant: \[ D = \begin{vmatrix} -4 & -4 & -4 \\ -4 & -4 & -4 \\ -4 & -4 & -4 \end{vmatrix} \] ### Step 3: Calculate the determinant Since all rows of the determinant are identical, the value of the determinant is: \[ D = 0 \] ### Conclusion Thus, the value of the determinant is: \[ \boxed{0} \]
To solve the problem, we need to find the value of 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} ...
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