- is an imaginary root of unity- i If a-b-c - 0- then a-b-c-23 - a-b-c-2-c-3 - 27abc-

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Standard XII

Mathematics

Applications of Dot Product

ω is an imagi...

Question

$ω$ is an imaginary root of unity.
(i) If $a + b + c = 0$ , then $(a + b ω + c ω^{2})^{3} + (a + b ω + c ω^{2} + c ω)^{3} = 27 a b c$ .

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Solution

$a + b + c = 0 ⟹ b + c = - a, c + a = - b$ and $a + b = - c$
Putting these value on the R.H.S. of result (i), we get $(a + b ω + c ω^{2})^{3} + (a + b ω^{2} + c ω)^{3} = 27 a b c$

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Similar questions

Q. State true or false:

ω

is an imaginary root of unity.

(a + b ω + c ω^{2})^{3} + (a + b ω^{2} + c ω)^{3} = (2 a - b - c) (2 b - a - c) (2 c - a - b)

Q. The value of

(a + b ω + c ω^{2})^{3} + (a + b ω^{2} + c ω)^{3}

is/are
[Here

ω

is a complex cube root of unity]

If $ω$ is a complex cube root of unity, then the value of $\frac{a + b ω + c ω^{2}}{c + a ω + b ω^{2}}$ + $\frac{a + b ω + c ω^{2}}{b + c ω + b ω^{2}}$ is :

Q. If cube root of unity is

1, ω, ω^{2}

then find the value of:

\frac{(a + b ω + c ω^{2})}{(c + a ω + b ω^{2})} + \frac{(a + b ω + c ω^{2})}{(b + c ω + a ω^{2})}

Q. If

x = a, y = b ω, z = c ω^{2}

, where

ω

is a complex cube root of unity, then

\frac{x}{a} + \frac{y}{b} + \frac{z}{c} =

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ω is an imaginary root of unity. (i) If a+b+c=0, then (a+bω+cω2)3+(a+bω+cω2+cω)3=27abc.

a+b+c=0⟹b+c=−a,c+a=−b and a+b=−cPutting these value on the R.H.S. of result (i), we get (a+bω+cω2)3+(a+bω2+cω)3=27abc

$ω$ is an imaginary root of unity.
(i) If $a + b + c = 0$ , then $(a + b ω + c ω^{2})^{3} + (a + b ω + c ω^{2} + c ω)^{3} = 27 a b c$ .

$a + b + c = 0 ⟹ b + c = - a, c + a = - b$ and $a + b = - c$
Putting these value on the R.H.S. of result (i), we get $(a + b ω + c ω^{2})^{3} + (a + b ω^{2} + c ω)^{3} = 27 a b c$