If the cube roots of unity be 1- w - w - w 2-then the roots of the equation x -13-8-0 are A- -1-1-2 - 1-2 -2B- -1-1-2 - 1-2 -2C- -1-1-1D- None of these

CONVENTIONAL APPROACH: (x 1)^3= 8⇒ x 1=( 8)^1/3 ⇒ x 1= 2, 2w,2w^2 ⇒ x= 1,1 2w,1 2w^2 Trick:By inspection, we see that (b) satisfies the equation i.e, ( 1 1)^3+8 ...

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Standard XII

Mathematics

Higher Order Equations

If the cube r...

Question

If the cube roots of unity be $1, w, w, w^{2}$ ,then the roots of the equation $(x - 1)^{3} + 8 = 0$ are

-1, 1+2ω,1+2ω²

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-1, 1-2ω,1-2ω²

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-1,-1,-1

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None of these

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Solution

The correct option is B -1, 1-2ω,1-2ω²
CONVENTIONAL APPROACH:
$(x - 1)^{3} = - 8 \Rightarrow x - 1 = (- 8)^{\frac{1}{3}}$
$\Rightarrow x - 1 = - 2, - 2 w, 2 w^{2} \Rightarrow x = - 1, 1 - 2 w, 1 - 2 w^{2}$
Trick:By inspection, we see that (b) satisfies the equation i.e,
( $- 1 - 1)^{3} + 8 - 0, (1 - 2 w - 1)^{3} + 8 - 0 (1 - 2 w^{2} - 1)^{2} + 8 - 0.$

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If the cube roots of unity be 1,w,w,w2 ,then the roots of the equation (x−1)3+8=0 are

The correct option is B -1, 1-2ω,1-2ω2CONVENTIONAL APPROACH: (x−1)3=−8⇒ x−1=(−8)13 ⇒x−1=−2,−2w,2w2 ⇒x=−1,1−2w,1−2w2 Trick:By inspection, we see that (b) satisfies the equation i.e, (−1−1)3+8−0,(1−2w−1)3+8−0(1−2w2−1)2+8−0.

If the cube roots of unity be $1, w, w, w^{2}$ ,then the roots of the equation $(x - 1)^{3} + 8 = 0$ are