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Let $a, b \in C, a \neq 0.$ Suppose $z_{1}, z_{2}, z_{3}$ are the roots of the equation ${(Z + a b)}^{3} = a^{3}$
Statement –I: Length of a side of the triangle with vertices $z_{1}, z_{2} a n d z_{3} i s \sqrt{3} | a |$
Statement-II: Roots of $z^{3} = a^{3}$ lie on a circle of radius $| a |$

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Statement I is true, Statement II true, statement II correct explanation for I

Statement I is true, Statement II is true and Statement II is not correct explanation for I

Statement I is true, Statement II is false

Statement I is false, Statement II is true

answer is B.

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Detailed Solution

Statement-II is true, as $z^{3} = a^{3} \Rightarrow {| z |}^{3} = {| a |}^{3} \Rightarrow | z | = | a |$

Next, ${(z + a b)}^{3} = a^{3} \Rightarrow z + a b = a, a w, a w^{2}$

$L e t z_{1} = a - a b, z_{2} = a w - a b a n d z_{3} = a w^{2} - a b$

$W e g a v e | z_{2} - z_{1} | = | a | | w - 1 | = \sqrt{3} a, | z_{3} - z_{2} | = \sqrt{3} a, | z_{1} - z_{3} | = \sqrt{3} a$

Thus, statement-I is also true but statement-II is not a correct explanation of statement-I

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