Let (1+x)3n=C0+C1x+C2x2+…+C3nx3n, and ω≠1 1 be a cube root of unityStatement-1: C0+C1ω+C2ω2+C3+C4ω+C5ω2+⋯=(−1)nStatement-2: Cube roots unity form a triangle of area 3 square units.

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Let $(1 + x)^{3 n} = C_{0} + C_{1} x + C_{2} x^{2} + \dots + C_{3 n} x^{3 n},$ and $ω \neq 1$ 1 be a cube root of unity

Statement-1: $C_{0} + C_{1} ω + C_{2} ω^{2} + C_{3} + C_{4} ω + C_{5} ω^{2} + \dots = (- 1)^{n}$

Statement-2: Cube roots unity form a triangle of area $\sqrt{3}$ square units.

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STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1

STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1

STATEMENT-1 is True, STATEMENT-2 is False

STATEMENT-1 is False, STATEMENT-2 is True

answer is C.

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Detailed Solution

$C_{0} + C_{1} ω + C_{2} ω^{2} + C_{3} + C_{4} ω + C_{5} ω^{2} + \dots$

$\begin{aligned} = \sum_{k = 0}^{3 n} C_{k} ω^{k} = (1 + ω)^{3 n} \\ = {(- ω^{2})}^{3 n} = (- 1)^{3 n} ω^{6 n} \\ = (- 1)^{n} (1) = (- 1)^{n} . \end{aligned}$

Statement-2 is false as area of triangle formed by cube roots of unity is

$3 \sqrt{3} / 4$ square units

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